Matrix equation Ax=b

Matrix equation Ax=b


If AA is an m×nm \times n matrix with columns a1a_1,…,ana_n, and if xx is in Rn\Bbb{R}^n, then the product of AA and xx is the linear combination of the columns in A using the corresponding entries in xx as weights. In other words,
linear combination of column

If we were to say that Ax=bAx=b, then basically:
a1x1++anxn=b a_1 x_1+\cdots+a_n x_n=b

which we see b is a linear combination of a1,,ana_1,\cdots,a_n. You will see questions where we have to solve for the entries of xx again, like last section.

We say that an equation in the form of Ax=bAx=b is a matrix equation.

Properties of AxAx
If AA is an m×nm \times n matrix, uu and vv are vectors in Rn\Bbb{R}^n, cc is a scalar, then:

1. A(u+v)=Au+AvA(u+v)=Au+Av
2. A(cu)=c(Au)A(cu)=c(Au)
  • 1.
    Matrix Equation Ax=b Overview:
    Interpreting and Calculating AxAx
    • Product of AA and xx
    • Multiplying a matrix and a vector
    • Relation to Linear combination

    Matrix Equation in the form Ax=bAx=b
    • Matrix equation form

    Solving x
    • Matrix equation to an augmented matrix
    • Solving for the variables

    Properties of Ax
    • Addition and subtraction property
    • Scalar property

  • 2.
    Computing Ax
    Compute the following. If it cannot be computed, explain why:
    computing Ax

    compute Ax

    calculating Ax

  • 3.
    Converting to Matrix Equation and Vector Equation
    Write the given systems of equations as a vector equation, and then to a matrix equation.
    6x1+2x23x3=1 6x_1+2x_2-3x_3=1
    2x15x2+x3=4 2x_1-5x_2+x_3=4
    x12x27x3=5 -x_1-2x_2-7x_3=5

  • 4.
    Solving the Equation AX=bAX=b
    Write the augmented matrix for the linear system that corresponds to the matrix equation Ax=bAx=b. Then solve the system and write the solution as a vector.
    Solving the Equation Ax=b

    solve Ax=b

  • 5.
    Ax=b with unknown b terms
    Let finding b in Ax=b and finding b in Ax=b. Show that the matrix equation Ax=bAx=b does have solutions for some bb, and no solution for some other bb’s.

  • 6.
    Understanding Properties of Ax
    Recall that the properties of the matrix-vector product Ax is:

    If AA is an m×nm \times n matrix, uu and vv are vectors in Rn\Bbb{R}^n, cc is a scalar, then:
    1. A(u+v)=Au+Av A(u+v)=Au+Av
    2. A(cu)=c(Au) A(cu)=c(Au)

    Using these properties, show that:
    A[(2u3v)(2u+3v)]=4A(u2)9A(v2) A[(2u-3v)(2u+3v)]=4A(u^2 )-9A(v^2)