113年 - 113 國立臺北大學_日間學士班轉學生招生考試試題_資訊工程學系日間學士班3年級：線性代數#147391

(a) If A and B are nonsingular n x n matrices, then A + B is also nonsingular and (A + B)⁻¹ = A⁻¹ + B⁻¹.

(b) If A and B are n x n matrices, then (A + B) (A - B) = A² - B².

(c) If x and y are two distinct vectors in ℝⁿ and Ax=Ay, then det(A)=0.

(d) If A and B are two equivalent matrices, then their determinants are equal.

(e) If x₁, x₂, ..., xₖ are vectors in a vector space V and Span(x₁, x₂, ..., xₖ) = Span(x₁, x₂, ..., xₖ₋₁), then x₁, x₂, ..., xₖ are linearly dependent.

(f) If A is an m x n matrix, then A and Aᵀ have the same nullity.

(g) Let L: ℝⁿ → ℝⁿ be a linear transformation. If L(x₁) = L(x₂), then the vectors x₁ and x₂ must be equal.

(h) If an m x n matrix A has linearly dependent columns and b is a vector in ℝᵐ, then b does not have a unique projection onto the column space of A.

(i) If Q is an orthogonal matrix, then Qᵀ also is an orthogonal matrix.

(j) If A is an n x n matrix, then A and Aᵀ have the same eigenvalues and same eigenvectors.

(a) Use elementary row operations to convert the above system into strictly triangular system. Then solve the system.

(b) Solve the linear system using Gaussian-Jordan Reduction.

(c) Solve the linear system using Cramer's Rule.

(d) Express the linear system in matrix representation Ax = b, then solve the linear system by computing A⁻¹.

(e) Compute the rank of A.

(f) Determine whether the vectors a₁, a₂, and a₃ are linearly independent.

(g) Determine the basis of the subspace N(A).

(h) Find the eigenvalues of A.

(i) For each eigenvalue, find the basis for the corresponding eigenspace.

(j) Compute A²⁰.

3. (20%) Let L(x) = (x₁ + x₂, x₂ - x₃)ᵀ be the transformation from ℝ³ to ℝ².
(a) Determine whether L is a linear transformation.(b) Determine the standard matrix representation A and find L(x) for x = [1, 2, 3]ᵀ.