Linear combination and vector equations RnR^n

Linear combination and vector equations Rn


Throughout the past lessons we have studied the notation of matrices, worked with systems of linear equations and learned how to represent them and solve them by either graphing or matrix row reduction. The time has come for us to learn about the usage of such notations and techniques when working with linear combinations and vectors, and for that reason, let us have a little introduction about vector notation in matrix form and the plane dimensions of such vectors before we enter into the main topic of today.
Recall a matrix is an array of numbers put in a rectangular bracket which dimensions are defined by how many rows and columns it has. So, a simple 3x3 matrix looks like this:

Linear combination and vector equations Rn
Equation 1: Example of a 3x3 matrix

A matrix with one column is called a column vector and it can be used in different algebraic operations along with other column vectors.

Linear combination and vector equations Rn
Equation 2: Example of a column vector in R2

The column vector is said to be in coordinate space R2R^{2} since it has two rows of real numbers, this means the vector has two dimensions and its graphic representation would lie on a two-dimensional plane. We can also have a vector in R3R^{3} which could be a three-dimensional vector such as the one below:

Linear combination and vector equations Rn
Equation 3: Example of a column vector in R3

You can actually have a column vector with as many rows as you want, and so, we say a vector is in the coordinate or vector space RnR^{n} where n will define how many rows are contained on it. We can summarize that a vector in RnR^{n} would have n entries in one column, such as:

Linear combination and vector equations Rn
Equation 4: Column vector in Rn

The amount of rows in a column vector is very important to denote the type of vectors you can operate with one another. Plainly said, column vectors can be added or subtracted with other column vectors only as long as they have the same amount of rows.

Linear combination and vector equations Rn
Equation 5: Examples of adding and subtracting column vectors

To add or subtract vectors take us to an important concept called the parallelogram rule of addition. The parallelogram rule of addition is used when working on the graphic representation of vectors, and it states that when adding two vectors we can represent them as the adjacent sides of a parallelogram drawn from the origin, which means that you are to move a vector parallely from its original position to have its tail start from the arrowhead of the other, and the resultant vector from such addition (or subtraction) would have the magnitude and direction resulting from this new vectors tail being in the origin (the starting point) and running until the arrowhead of the outermost vector. The graphic representation of the addition of two vectors is shown below:

Linear combination and vector equations Rn
Figure 1: Graphing representation of adding vectors

We can also multiply a column vector by a scalar:

Linear combination and vector equations Rn
Equation 6: Multiplying a scalar to a column vector

And so, to operate (add, subtract, or multiply a scalar) on vectors, here are the following properties of RnR^{n}, where uu, vv, ww are vectors and cc, dd are scalars:

Linear combination and vector equations Rn
Equation 7: Properties of vector operations for vectors in Rn

What is a linear combination


Given vectors v1v_{1}, ...,vpv_{p} in RnR^{n} with scalars c1c_{1}, ... ,cpc_{p}, the vector equation xx is defined as:

x=v1c1+v2c2+...+vpcpx=v_{1}c_{1} + v_{2}c_{2} + ... + v_{p} c_{p}
Equation 8: Linear combination definition

Where x is a linear combination of vectors v1v_{1}, ...,vpv_{p} with scalars c1c_{1}, ... ,cpc_{p}.
The linear combinations of v1v_{1}, ...,vpv_{p} is the same as saying this is a linear combination span: Span{v1v_{1}, ...,vpv_{p}}

There is something very important to note in this kind of notation before we continue on the section teaching about vector equations. For the linear combination definition we just saw above, the range of v1v_{1}, ...,vpv_{p} refers to the range of column vectors and the scalars are by definition magnitude values because we usually refer to this as a value of one (this is always the case for linear combinations which would result in linear systems of equations). The reason to use a value of one for the scalars comes from the fact that we will be converting such linear combinations into vector equations, which later can be transformed into augmented matrices which by now, we know can be re-written as a system of equations.

Therefore, by the time such transformations (or transcriptions) of such linear combinations to a system of linear equations finally occur, we will see how such scalars hereby named c1c_{1}, ... ,cpc_{p} actually represent the variables in the equation which provide the dimensions for the graphic representation of such systems of linear combinations.

At the moment this all may seem rather confusing, so let us get into the next section where we will explain step by step the method of going from a linear combination notation such as the one shown in the definition in equation 8 to the algebraic notation representing a system of linear equations, passing through vector equations and augmented matrices in the process.

What is a vector equation



Equation 8 has so far been called the definition of the linear combination, in truth, this is also a vector equation, so it is now time for us to take it as such and expand it. How to do it? Actually we have already hinted in the past section it is all about linear systems of equations, so now, let us represent a system of linear equations as the vector equation from the definition in equation 8:

Linear combination and vector equations Rn
Equation 9: A system of linear equations as a vector equation

Notice what has been done here. We took a system of linear equations which can be found in the left-most side above and wrote them as a vector equation notation by separating the terms with distinct variables and finally (in the right-most part), provide a vector equation in which you can clearly see the values corresponding to the two ranges of terms defined by v1v_{1}, ...,vpv_{p} and c1c_{1}, ... ,cpc_{p}. Remember that v1v_{1}, ...,vpv_{p} refers to column vectors, and c1c_{1}, ... ,cpc_{p} as scalars, so:

Linear combination and vector equations Rn
Equation 10: A system of linear equations as a vector equation

The vector equation resulting from the linear system of equations is:

Linear combination and vector equations Rn
Equation 11: Vector equation from system of linear equations

Thus we can say a vector equation such as the one shown in equation 11 is a linear combination if its left hand side is equal to its right hand side, which can only happen when the scalars x1,x2,x3 are all equal to one.

To elaborate on this explanation and make the relationship between a linear system of equations and a linear combination clearer, let us remember what we had mentioned before: the array v1v_{1}, ... ,vpv_{p} refers to the column vectors, while c1c_{1}, ... ,cpc_{p} end up referring to the variables of the system of equations, in this case x1,x2,x3x_{1}, x_{2}, x_{3}. This is the confusion we were trying to clear out in the last section, why do we call x1,x2,x3x_{1}, x_{2}, x_{3} scalars if we know that when transcribing them from a system of equations, they happen to be the variables? Because they have a value of 1. Since we are ONLY talking about systems of linear equations, by definition, these variables would have a magnitude of one since this is the only value that allows the left hand side and right hand side of equation 11 to be equal.
In other words, the system of linear equations in the left-most side of equation 9 and equation 10 could be very well written as:

Linear combination and vector equations Rn
Equation 12: System of linear equations using variable names x, y and z.

How do we know that? Well it so happens that if you have a vector equation, similar to the one found in equation 11, we can write it as an augmented matrix. Take the example below:

Linear combination and vector equations Rn
Equation 13: Vector equation to augmented matrix

Which, as we know from our lesson on representing a linear system as a matrix, an augmented matrix notation follows the next rules:

Linear combination and vector equations Rn
Figure 2: General diagram for constructing an augmented matrix from a system of linear equations

Which means that variable 1, variable 2 and variable 3 are the same as x1,x2,x3x_{1}, x_{2}, x_{3} or just xx, yy and zz, it really does not matter which name you give to them! For this lesson we will be looking into the linear combination of normal random variables.

Linear combination and vector equations Rn
Equation 14: Augmented matrix from equation 13

Once we have the augmented matrix we know how to solve it! Remember from our lessons on solving a linear system with matrices using gaussian elimination and rrow reduction and echelon formswe learned the technique to use the three types of matrix row operations in order to solve a system of linear equations. So, if you have any doubts on how row reduce the matrix shown above in equation 14, we recommend you to go back and do a little review of the lessons suggested here. Also, do not forget to check out the videos accompanying this lesson, since all of our operations have been shown in there.

Linear combination method examples


As the title of this lesson describes, it is time now we work on exercises which represent linear combination examples. Notice we will go advancing step by step throughout the exercises, from how to write vector equations to how to find a vector linear combination, which can be found in the last problems which we will solve by linear combination.

Example 1

Consider the two vectors shown below and then compute the operations in parts a), b) and c):

Linear combination and vector equations Rn
Equation 15: Vectors u and v

Compute

a) \quad u+2vu + 2v

Linear combination and vector equations Rn
Equation 16: Computing u+2v

b) \quad 2uv2u - v

Linear combination and vector equations Rn
Equation 17: Computing 2u-v

c) \quad 5u+0v5u + 0v

Linear combination and vector equations Rn
Equation 18: Computing 5u+0v


Example 2

Using the next system of equations, write it as a vector equation:

Linear combination and vector equations Rn
Equation 19: System of linear equations

In order to write the linear system as a vector equation, we start by separating all of the terms containing the same variable as columns, and then rewrite them as such:

Linear combination and vector equations Rn

Linear combination and vector equations Rn
Equation 20: From algebraic system of linear equations to vector equation form

As you can see in equation 20, the right-most equation is the one consisting of the vector equation form in which the variables x1x_{1}, x2x_{2} and x3x_{3} are multiplying a column vector comprised of the coefficients found in the system for each variable.

Example 3

Write the given vector equation as a system of equations:

Linear combination and vector equations Rn
Equation 21: vector equation

We just easily distribute the coefficients to each of the variables:

Linear combination and vector equations Rn
Equation 22: Transforming the vector equation into a system of linear equations


***

For the next example, we will be using the linear combination method in order to find a vector parametric equation with explicit values, which will allow us to confirm if the given elements are linear combination vectors.

Example 4


a) \quad Determine by linear combinations method if bb is a linear combination of a1a_{1}, a2a_{2} when:

Linear combination and vector equations Rn
Equation 23: Column vectors to use

For bb to be a linear combination with the given column vectors a1a_{1} and a2a_{2} are added, it means that the vector equation you can construct with these terms will have equal values on its left hand side and on its right hand side. Therefore, we start by writing the vector equation from the given column vectors and convert it into the augmented matrix:

Linear combination and vector equations Rn
Equation 24: Making the vector equation and the augmented matrix

What we need to do here is to find the values of x1x_{1} and x2x_{2} by computing the reduced echelon form of the augmented matrix using row reduction. If we can obtain a particular value for each, then it means b is a linear combination of a1a_{1}, a2a_{2}, and so, we row reduce to solve for the unknown variables:

Linear combination and vector equations Rn

Linear combination and vector equations Rn
Equation 25: Values for which b is a linear combination of a1 and a2

Since we were able to solve for the variables x1x_{1} and x2x_{2} without and issue, then we can see that b will be a linear combination for a1a_{1} and a1a_{1} when:

x1=344=8.5x_{1}= \frac{34}{4} = 8.5 and x2=8x_{2} = 8
Equation 26: Final solution for the values of the variables x1 and x2

b) \quad Determine if bb is a linear combination of a1a_{1}, a2a_{2}, and a3a_{3} when:


Linear combination and vector equations Rn
Equation 27: Column vectors to use

Again, we write down the vector equation, transform it into the augmented matrix and then row reduce it to find the values of the unknown variables:

Linear combination and vector equations Rn
Equation 28: Vector equation and augmented matrix

Linear combination and vector equations Rn


Linear combination and vector equations Rn

Linear combination and vector equations Rn

Linear combination and vector equations Rn
Equation 29: Row reducing to echelon form to solve variables

And as you can see, since we could find the value of all the variables, then b is a nontrivial linear combination of a1a_{1} and a1a_{1}.

***

We finalize the lesson here after having done linear combination of matrices examples. Before we pass into the next lesson, we recommend you to take a look into these lessons on linear combinations and vector equations, where you will find even more examples and also, some graphic representations of the equations and systems we have seen today.

Linear combination and vector equations RnR^n

Lessons

A matrix with one column is called a column vector. They can be added or subtracted with other column vectors as long as they have the same amount of rows.

Parallelogram Rule for Addition: if you have two vectors uu and vv, then u+vu+v would be the fourth vertex of a parallelogram whose other vertices are u,(0,0)u,(0,0),and vv

Here are the following algebraic properties of Rn\Bbb{R}^n
1. u+v=v+uu+v=v+u
2. (u+v)+w=u+(v+w)(u+v)+w=u+(v+w)
3. u+0=0+u=uu+0=0+u=u
4. u+(u)=u+u=0u+(-u)=-u+u=0
5. c(u+v)=cu+cvc(u+v)=cu+cv
6. (c+d)u=cu+du(c+d)u=cu+du
7. c(du)=(cd)(u)c(du)=(cd)(u)
8. 1u=u1u=u

Given vectors v1,,vpv_1,\cdots,v_p in Rn\Bbb{R}^n with scalars c1,,cpc_1,\cdots,c_p, the vector xx is defined by

x=v1c1++vpcpx=v_1 c_1+\cdots+v_p c_p

Where xx is a linear combination of v1,,vpv_1,\cdots,v_p.

The linear combinations of v1,,vpv_1,\cdots,v_p is the same as saying Span{v1,,vpv_1,\cdots,v_p}.
  • Introduction
    Vector Equations in Rn\Bbb{R}^n Overview:
    a)
    Vectors in R2\Bbb{R}^2
    • Column vectors with 2 rows
    • Adding, subtracting, and multiplying 2D vectors
    • Graphing vectors in 2D
    • Parallelogram Rule for Addition

    b)
    Vectors in R3\Bbb{R}^3
    • Column vectors with 3 rows
    • Adding, subtracting, and multiplying 3D vectors
    • Graphing vectors in 3D

    c)
    Vectors in Rn\Bbb{R}^n
    • Column vector with nn rows
    • Algebraic properties

    d)
    Linear Combinations and Spans
    • Vectors and weights
    • Vector equations
    • Finding a linear combination with row reduction


  • 1.
    Calculating Vectors in Rn\Bbb{R}^n
    Consider the two vectors Calculating vectors in R^n, vector 1, and Calculating vectors in R^n, vector 2. Compute:
    a)
    u+2v u+2v

    b)
    2uv 2u-v

    c)
    5u+0v 5u+0v


  • 2.
    Converting Systems Of Equations And Vector Equations
    Write the given systems of equations as a vector equation.

    2x1+x25x3=42x_1+x_2-5x_3=4
    x1+3x2+2x3=1x_1+3x_2+2x_3=1
    4x1x28x3=2-4x_1-x_2-8x_3=-2

  • 3.
    Write the given vector equation has a system of equations
    vector equation and system of equations

  • 4.
    Linear Combinations with Known terms
    Determine if bb is a linear combination of a1a_1, a2a_2 in part a. Determine if bb is a linear combination of a1a_1, a2a_2, and a3a_3 in part b and c.
    a)
    linear combination

    b)
    linear combination and vectors

    c)
    linear combination with three vectors


  • 5.
    Linear Combinations with Unknown terms
    For what value(s) of kk is bb in the plane spanned by a1a_1 and a2a_2 if:
    linear combination with unknown terms