Linear combination and vector equations RnR^n

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}.
  • 1.
    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


  • 2.
    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


  • 3.
    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

  • 4.
    Write the given vector equation has a system of equations
    vector equation and system of equations

  • 5.
    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


  • 6.
    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