Question

Are the following statements true or false?

1. For an (m × n) matrix (A), vectors in the null space of (A) are orthogonal to vectors in the row space of (A).

a) True

b) False

2. (A^TA = I).

a) True

b) False

3. For any scalar (c), (cA = A).

a) True

b) False

4. If (u) and (v) are nonzero vectors and (u ⋅ v = 0), then (u) and (v) are orthogonal.

a) True

b) False

5. If (v) is orthogonal to every vector in a subspace (W), then (v) is in (W).

a) True

b) False

Answer 1

1. True, 2. False, 3. False, 4. True, 5. False

Step-by-step explanation:

1. For an (m × n) matrix (A), vectors in the null space of (A) are orthogonal to vectors in the row space of (A). This statement is true. The null space of a matrix consists of vectors that get mapped to the zero vector when multiplied by the matrix. These vectors are orthogonal to the row space of the matrix, which consists of the linear combinations of the row vectors of the matrix.

2. (A^TA = I). This statement is false. The product of a matrix and its transpose does not necessarily give the identity matrix. It only gives the identity matrix if the matrix is invertible.

3. For any scalar (c), (cA = A). This statement is false. When multiplying a matrix by a scalar, the scalar multiplies every element of the matrix, not just the scalar itself.

4. If (u) and (v) are nonzero vectors and (u ⋅ v = 0), then (u) and (v) are orthogonal. This statement is true. The dot product of two vectors is zero if and only if the vectors are orthogonal to each other.

5. If (v) is orthogonal to every vector in a subspace (W), then (v) is in (W). This statement is false. A vector can be orthogonal to all vectors in a subspace without being an element of that subspace.