Questions: From the following, select all statements that are true. There may be more than one correct answer. A. If x is not in a subspace W, then x-projw(x) is not zero. B. In a QR factorization, say A=QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A. C. If W=Spanv1, v2, v3, and if v1, v2, v3 is an orthogonal set in W, then v1, v2, v3 is an orthonormal basis for W. D. None of the above.

From the following, select all statements that are true. There may be more than one correct answer.
A. If x is not in a subspace W, then x-projw(x) is not zero.
B. In a QR factorization, say A=QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A.
C. If W=Spanv1, v2, v3, and if v1, v2, v3 is an orthogonal set in W, then v1, v2, v3 is an orthonormal basis for W.
D. None of the above.

Transcript text: From the following, select all statements that are true. There may be more than one correct answer. A. If $\mathbf{x}$ is not in a subspace $W$, then $\mathbf{x}-\operatorname{proj}_{w}(\mathbf{x})$ is not zero. B. In a $Q R$ factorization, say $A=Q R$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$. C. If $W=\operatorname{Span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{\mathbf{3}}\right\}$, and if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is an orthogonal set in $W$, then $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{\mathbf{3}}\right\}$ is an orthonormal basis for $W$. D. None of the above. Submit answer Next item

Solution

Solution Steps

To determine which statements are true, we need to analyze each statement based on linear algebra principles:

A. If $\mathbf{x}$ is not in a subspace $W$, then $\mathbf{x} - \operatorname{proj}_{W}(\mathbf{x})$ is not zero.

This statement is true because the projection of $\mathbf{x}$ onto $W$ is the closest point in $W$ to $\mathbf{x}$. If $\mathbf{x}$ is not in $W$, the difference $\mathbf{x} - \operatorname{proj}_{W}(\mathbf{x})$ will be non-zero.

B. In a $QR$ factorization, say $A = QR$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$.

This statement is true because in $QR$ factorization, $Q$ is an orthogonal matrix whose columns are orthonormal vectors that span the column space of $A$.

C. If $W = \operatorname{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$, and if $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is an orthogonal set in $W$, then $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is an orthonormal basis for $W$.

This statement is false because an orthogonal set is not necessarily orthonormal. For it to be orthonormal, each vector must also be of unit length.

D. None of the above.

This statement is false because statements A and B are true.

Solution Approach

Analyze each statement based on linear algebra principles.
Determine the truth value of each statement.

Step 1: Analyze Statement A

Statement A: If $\mathbf{x}$ is not in a subspace $W$, then $\mathbf{x} - \operatorname{proj}_{W}(\mathbf{x})$ is not zero.

This statement is true because the projection of $\mathbf{x}$ onto $W$ is the closest point in $W$ to $\mathbf{x}$. If $\mathbf{x}$ is not in $W$, the difference $\mathbf{x} - \operatorname{proj}_{W}(\mathbf{x})$ will be non-zero.

Step 2: Analyze Statement B

Statement B: In a $QR$ factorization, say $A = QR$ (when $A$ has linearly independent columns), the columns of $Q$ form an orthonormal basis for the column space of $A$.

This statement is true because in $QR$ factorization, $Q$ is an orthogonal matrix whose columns are orthonormal vectors that span the column space of $A$.

Step 3: Analyze Statement C

Statement C: If $W = \operatorname{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$, and if $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is an orthogonal set in $W$, then $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is an orthonormal basis for $W$.