Complete each of the following sentences with MUST, MIGHT, or CANNOT. (a) Let A be a square matrix. If Ax = Ay for distinct x and y, then A _ be invertible. (b) Let T be a matrix transformation with standard matrix A. The kernel of T _ equal the null space of A. The range of T – equal the column space of A. (c) Let S = {u, v} be a set of vectors. If w is in Span{S}, then w be in S. (d) Let A be a non-zero, non-invertible 3 x 3 matrix. If the column space of A does not form a line, then the null space of A form a line.
The Correct Answer and Explanation is :
Let’s complete the sentences with MUST, MIGHT, or CANNOT and provide explanations:
(a) Let A be a square matrix. If (Ax = Ay) for distinct (x) and (y), then A CANNOT be invertible.
Explanation:
If (Ax = Ay) for distinct (x) and (y), it means that the linear transformation represented by (A) maps two distinct vectors (x) and (y) to the same vector. This violates the property of injectivity required for invertibility. A matrix is invertible if and only if its transformation is injective (no two distinct vectors map to the same output). Since distinct vectors (x) and (y) map to the same vector, the matrix (A) is not injective and therefore CANNOT be invertible.
(b) Let T be a matrix transformation with standard matrix A. The kernel of T MUST equal the null space of A. The range of T MUST equal the column space of A.
Explanation:
The kernel of a matrix transformation (T) is the set of all vectors (v) such that (T(v) = 0), which is equivalent to the null space of the matrix (A). The null space of (A) consists of all vectors (x) such that (Ax = 0), which is the definition of the kernel of (T). Similarly, the range of (T) corresponds to the set of all possible outputs of (T), which is the same as the column space of the matrix (A). These relationships are fundamental to linear transformations and matrix theory.
(c) Let S = {u, v} be a set of vectors. If w is in Span{S}, then w MIGHT be in S.
Explanation:
The span of (S = {u, v}) is the set of all linear combinations of (u) and (v). This means that (w) can be written as a linear combination of (u) and (v). However, (w) is not necessarily in (S), because (w) could be any combination of (u) and (v), not just (u) or (v) themselves. Therefore, it is possible that (w) is not one of the original vectors in the set (S), but it might be if it is exactly equal to (u) or (v).
(d) Let A be a non-zero, non-invertible 3 x 3 matrix. If the column space of A does not form a line, then the null space of A MUST form a line.
Explanation:
For a 3×3 matrix (A) that is non-zero and non-invertible, its column space must be a subspace of (\mathbb{R}^3). If the column space does not form a line, it suggests that the column space is at least two-dimensional (in other words, it spans a plane). Since (A) is non-invertible, the rank of (A) is less than 3. A 3×3 matrix with a rank of 2 (indicating a column space that spans a plane) will have a null space that is one-dimensional, meaning it forms a line. Therefore, the null space MUST form a line.