Question

1. (3 points) (Orthonormal Bases/Gram-Schmidt) Consider the matrix $A = \begin{bmatrix} \vec{a}_1 & \vec{a}_3 & \vec{a}_1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 4 \\ 0 & 0 & 5 \\ 0 & 3 & 6 \end{bmatrix}$. (a) Find an orthonormal basis $\{\vec{q}_1, \vec{q}_2, \vec{q}_3\}$ for the columns of the matrix A. (b) Find the matrix R for which A = QR, i.e., the matrix which expresses the vectors $\vec{a}_1, \vec{a}_2$ and $\vec{a}_3$ as linear combinations of the orthonormal basis vectors. (Hint: what is the inverse of Q, as discussed in class?). (c) From the result above, find the constants $c_{11}, c_{12}$ and $c_{13}$ for which $\vec{a}_1 = c_{11}\vec{q}_1 + c_{12}\vec{q}_2 + c_{13}\vec{q}_3.$

          1. (3 points) (Orthonormal Bases/Gram-Schmidt) Consider the matrix
$A = \begin{bmatrix} \vec{a}_1 & \vec{a}_3 & \vec{a}_1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 4 \\ 0 & 0 & 5 \\ 0 & 3 & 6 \end{bmatrix}$.
(a) Find an orthonormal basis \(\{\vec{q}_1, \vec{q}_2, \vec{q}_3\}\) for the columns of the matrix A.
(b) Find the matrix R for which A = QR, i.e., the matrix which expresses the vectors
$\vec{a}_1, \vec{a}_2$ and $\vec{a}_3$ as linear combinations of the orthonormal basis vectors. (Hint: what
is the inverse of Q, as discussed in class?).
(c) From the result above, find the constants $c_{11}, c_{12}$ and $c_{13}$ for which
$\vec{a}_1 = c_{11}\vec{q}_1 + c_{12}\vec{q}_2 + c_{13}\vec{q}_3.$

1. (3 points) (Orthonormal Bases/Gram-Schmidt) Consider the matrix
A =
< b m a t r i x >
=
< b m a t r i x >.
(a) Find an orthonormal basis {q⃗1, q⃗2, q⃗3} for the columns of the matrix A.
(b) Find the matrix R for which A = QR, i.e., the matrix which expresses the vectors
a⃗1, a⃗2 and a⃗3 as linear combinations of the orthonormal basis vectors. (Hint: what
is the inverse of Q, as discussed in class?).
(c) From the result above, find the constants c11, c12 and c13 for which
a⃗1 = c11q⃗1 + c12q⃗2 + c13q⃗3.

Added by Sheri Z.

Question

Please give Ace some feedback