Problem 1

(a). Let Be the Composition of Linear Transformations on that First Rotate Counterclockwise about the Origin by 90 Degree, and then Reflect about the line . Find Its Matrix

(b). Determine if is Invertible, if Not, Explain Why, if So, Compute Its Inverse

is invertible as it's determinant is nonzero

Problem 2

Indicate if each of the following is a linear subspace. Explain why if it is not. No justification needed if it is.

(a). All the Vectors with

No it doesn't as the zero vector is not included in the set.

(b). The Intersection of Two Subspaces of

Yes, since for every vector and must be in both subspaces and addition of those two vectors must also be in both subspaces as it could be rewritten as two vectors in one of the subspaces. The other properties come from the definition of a subspace and are equivalent for the intersection.

(c). The Union of the -axis and the -axis on

No because the vector and are in the set but is not so it's not closed under vector addition.

Problem 3

Let be an matrix with column vectors , , , . For each of the following statements say if they are true or false and justify your answer.

(a). If is Linearly Independent and , then is a Linear Combination of

True, if was not a linear combination of the previous elements via the invertible matrix theorem.

(b). There Exists a Vector Which is not in the Span of

True, there needs to be at least vectors to span

(c). If is Linearly Independent then there Exists a Vector such that the Determinant of the Matrix Obtained from Replacing the Last Column by Has Nonzero Determinant

True, because has elements and thus given a which is linearly independent to the others we can get a nonzero determinant.

Problem 4

Let be the matrix whose columns are the vectors and in . Suppose the reduced echelon form of A is

(a). What is the Rank of ? Explain Briefly

The rank of is 3 as there are 3 pivot positions in the row echelon form.

(b). What is the Dimension of the Null Space of ? Explain Briefly

Via the rank theorem the dimensions of the null space are .

(c). Give a Basis for the Null Space of

(d). The Vector is in the Null Space of . Find It's Coordinate Vector in the Basis You Give in part (c)

Problem 5

(a). Find the Determinant of the Matrix

(b). Find the Basis for the Eigenspace of the Matrix Corresponding to the Eigenvalue

The basis for the eigenspace of formed with is the set and

Problem 6

Answer true or false for the following questions

(a). For Matrices , We Have

False, consider two matrices and

we can see this forms terms like which don't show up on the other side of the question .

(b). It is Possible that a Matrix Has Rank 6

False it is not possible as it doesn't have enough rows to have 6 pivot points.

(c). If , , and Are Square Matrices, and is Invertible, then

False the order is incorrect as

(d). For a Matrix and a Scalar We Have

False because multiplies all row by and the determinant gets scaled by a factor of for every row that is scaled by so

(e). A Square Matrix with Entirely Zeros in the Diagonal is Never Invertible

False consider it's determinant is -1 and it's therefore invertible.

(f). Every Spanning Subset of Contains a Basis for

True the spanning subset must contain the basis.