MTH501 Quiz 4 January 2026

Question:

If {u1,u2,…,up}{u1,u2,…,up}\{u_{1}, u_{2}, …, u_{p}\} is an orthonormal basis for a subspace WWW of RnRnR^n, then ...

A. Projwy=(y.u1)up+(y.u2)up+…+(y.up)upProjwy=(y.u1)up+(y.u2)up+…+(y.up)upProj_{w}y=(y.u_{1})u_{p}+(y.u _{2})u_{p}+…+(y.u_{p})u_{p} B. Projwy=(y.u1)u1+(y.u2)u2+…+(y.up)upProjwy=(y.u1)u1+(y.u2)u2+…+(y.up)upProj_{w}y=(y.u_{1})u_{1}+(y.u _{2})u_{2}+…+(y.u_{p})u_{p} Copied: 1/8/2026, 10:01:49 PM

Question:

Two vectors are _______if at least one of the vector is a multiple of the other

A. linearly independent

B. linearly dependent

Copied: 1/8/2026, 10:03:05 PM

Question:

Two vectors uuu and vvv in RnRnR^n are orthogonal if ------

A. u.v=0u.v=0u.v=0 �B�.� �u�.�v"`�0�u�.�v"`�0�u�.�v� �\�n�e�q� �0 Copied: 1/8/2026, 10:03:47 PM

Question:

A unit vector is obtained by ------

A. $u={||v||}/{||v||}

B. $u=v/{||v||}

C. None of the above D. $u={||v||}/v Copied: 1/8/2026, 10:05:55 PM

Question:

Let W be a subspace of RnRn{R^n} and {u1,u2,...,up}{u1,u2,...,up}\{ {u_1},{u_2},...,{u_p}\} is any orthogonal basis of W, then y\^=c1u1+c2u2+...+cnuny\^=c1u1+c2u2+...+cnun\mathop y\limits^\^ = {c_1}{u_1} + {c_2}{u_2} + ... + {c_n}{u_n} Where

A. cj=y.ujuj.ujcj=y.ujuj.uj{c_j} = \frac{{y.{u_j}}}{{{u_j}.{u_j}}} B. cj=y.ujycj=y.ujy{c_j} = \frac{{y.{u_j}}}{y} C. None of these D. cj=y.ujujcj=y.ujuj{c_j} = \frac{{y.{u_j}}}{{u_j}} Copied: 1/8/2026, 10:06:55 PM

Question:

Elementary row operations on a matrix do not affect the -------- relations among the columns of the matrix. A. Linear independence

B. Linear dependence

Copied: 1/8/2026, 10:07:35 PM

Question:

The matrix A^T(Transpose of A) x A is ________ if and only if the columns of A are linearly independent.

A. singular B. scalar C. symmetric

D. invertible

Copied: 1/8/2026, 10:22:36 PM

Question:

If a matrix U has orthonormal columns, then UUT (transpose of U) = ________.

A. negative

B. I

C. 0 D. positive Copied: 1/8/2026, 10:08:55 PM