Problem 1

Let and

(a). Find

(b). Find the Inverse Matrix of

(c). Solve

Problem 2

The matrix of a linear transformation is

(a). Find a Basis for

(b). Find a Basis for the Range of

Problem 3

(a). Explain why is a Basis of , , , and

, and and are linearly independent so they span , we can prove that and are linearly independent as there exists constants and such that

(b). Find the Determinant of the Matrix for a Variable

Problem 4

(a). Find the Eigenspace of the Matrix for Eigenvalue

The eigenspace of with is the span of

(b). Find the Eigenvalues of the Matrix

Problem 5

Are the following statements true or false.

(a). If and Are Invertible Matrices then is Invertible

True, Given two invertible matrices and if both are invertible their product is also invertible, we can thus conclude is invertible and is also invertible.

(b). Let , Denote Square Matrices, then

False, this equation results in and since for all matrices and the original equation does not hold.

(c). Any Set of 5 Nonzero Vectors in is Linearly Dependent

True, the basis in can only have 4 elements at a maximum any more are redundant and thus linearly dependent.

(d). If the Null Space of a Matrix a is 2 Dimensional, then the Column Space of a is 4 Dimensional

True, via the rank theorem .

(e). The -cofactor of a Matrix is the Matrix Obtained by Deleting from the Row and Column

False the cofactor is the matrix minor (the determinant of the result of after deleting the row and column) times .

(f). If , then Two Rows or Two Columns Are Proportional, or a Row or a Column is Zero

False consider the matrix with a determinant 0 .