1 - Intro to vectors & matrices

Question 1

In the equation y = Ax, y is a linear combination of the ________ of A, where the x’s are the _______.

Answer 1

In the equation y = Ax, y is a linear combination ​ of the columns of A, where the x’s are the coefficients (that scale the column vectors).

Question 2

A matrix-vector product corresponds to the abstract notion of a _____ _________, which takes n-vectors as inputs and produces m-vectors as outputs.

So y = Ax,

y = ____

Answer 2

A matrix-vector product corresponds to the abstract notion of a linear transformation, which takes n-vectors as inputs and produces m-vectors as outputs.

So y = Ax,

y = T_A(x)

Question 3

A linear transformation or matrix has rows that correspond to the inputs/outputs, and columns that correspond to the inputs/outputs.

Answer 3

A linear transformation or matrix has rows that correspond to the outputs, and columns that correspond to the inputs.

(Think of the matrix as a machine that eats material from the top and spits it out on the side)

Question 4

When a matrix is square and ______ then there exists an _____ matrix ___ that undoes the effect of matrix A.

Answer 4

When a matrix is square and invertible then there exists an inverse matrix A^-1 that undoes the effect of matrix A.

Thus, A^-1(A(x)) = x