Linear Algebra

2. Fill in the blank:

A linear operator T is called _______ if T = T^*, where T^* is the adjoint of T.A. Tensor Product

B. Self-adjoint

C. Eigenvalues

D. Scalar Multiplication

3. Find det C for any value of x. You could use linearity in row 1.A. det C = x + 1. The calculation of the determinant is incorrect.

B. det C = 0. The determinant is not zero unless x = 1.

C. det C = x. The determinant is not simply the value of x.

D. det C = 1 - x or det C = xC₁₁ + det B = −x + 1. Check this answer (zero), for x = 1 when C = A.

4. Fill in the blank:A vector space over a field F is a set V equipped with two operations: vector addition and _______, satisfying certain axioms.A. Tensor Product

B. Self-adjoint

C. Eigenvalues

D. Scalar Multiplication

5.  This problem is to find the determinants ofFind det A and give a reason. A. det A = 1 because it has a diagonal of 1s

B. det A = 4 because there are four 1s in the first row

C. det A =0 because two rows are equal

D. det A = 1 because it's a 4x4 matrix

6. Find the cofactor C₁₁ and then find det B. This is the volume of what region in R⁴A. The cofactor C₁₁​ = 1 and det B = 2. This is the area of a plane in R⁴.

B. The cofactor C₁₁ = −1 and det B = 1. This is the volume of a box in R⁴ with edges given by the rows of BC. The cofactor C₁₁​ = 0 and det B = -1. This is the volume of a sphere in R⁴.

D. The cofactor C₁₁ ​= −1 and det B = 0. This is the volume of a cube in R⁴.