Question

In Exercises 17 and 18, all vectors and subspaces are in \({\mathbb{R}^n}\). Mark each statement True or False. Justify each answer.

a. If \(W = {\rm{span}}\left\{ {{x_1},{x_2},{x_3}} \right\}\) with \({x_1},{x_2},{x_3}\) linearly independent,

and if \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is an orthogonal set in \(W\) , then \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) is a basis for \(W\) .

b. If \(x\) is not in a subspace \(W\) , then \(x - {\rm{pro}}{{\rm{j}}_W}x\) is not zero.

c. 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 \(A\).

Short answer

a. False, because all \({v_i}\)’s should be non-zero to form a basis.

b. True, because \({\rm{pro}}{{\rm{j}}_W}x \in W\).

c. True, from the definition of \(QR\) factorization.

Step-by-step solution

 Step 1: \(QR\) factorization of a Matrix

A matrix with order \(m \times n\) can be written as the multiplication of an upper triangular matrix \(R\) and a matrix \(Q\) which is formed by applying the Gram–Schmidt orthogonalization processto the \({\rm{col}}\left( A \right)\).

The matrix \(R\) can be found by the formula \({Q^T}A = R\).

Checking whether the given statements are true of false

a.

Let \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) be a orthogonal set. Again let \({v_1} = 0\) then \(\left\{ {0,{v_2},{v_3}} \right\}\) will be still a orthogonal set. But this will not be a basis of the set \(W\).

So, the given statement is false, as nothing is given about non-zero vectors to form basis.

b.

The given statement is true. Since \(x\) is not a subspace of \(W\), then the projection of \(x\) in \(W\), that is \({\rm{pro}}{{\rm{j}}_W}x\) will not be equal. Because \({\rm{pro}}{{\rm{j}}_W}x \in W\).

c.

If \(A\) has a \(QR\) factorization that is \(A = QR\) then the columns of the matrix \(A\) are the vectors of the orthonormal basis of \({\rm{Col}}A\), so the given statement is true.