MATH347: Linear algebra for applications
This assignment is a worksheet of exercises intended as preparation for the Final Examination. You should:
Review Lessons 1 to 12
Set aside 60 minutes to solve these exercises. Each exercise is meant to be solved within 3 minutes. If you cannot find a solution within 3 minutes, skip to the next one.
Check your answers in Matlab. Revisit theory for skipped or incorrectly answered exercies.
Turn in a PDF with your brief handwritten answers that specify your motivation, approach, calculations, answer. It is good practice to start all answers by briefly recounting the applicable definitions.
Find the linear combination of vectors \(\boldsymbol{u}= \left[ \begin{array}{lll} 1 & 1 & 1 \end{array} \right]\), \(\boldsymbol{v}= \left[ \begin{array}{lll} 1 & 2 & 3 \end{array} \right]\) with scaling coefficients \(\alpha = 2\), \(\beta = 1\).
Express the above linear combination \(\boldsymbol{b}\) as a matrix-vector product \(\boldsymbol{b}=\boldsymbol{A}\boldsymbol{x}\). Define \(\boldsymbol{x}\) and the column vectors of \(\boldsymbol{A}= \left[ \begin{array}{ll} \boldsymbol{a}_1 & \boldsymbol{a}_2 \end{array} \right]\).
Consider \(\boldsymbol{u}= \left[ \begin{array}{lll} 1 & 1 & 0 \end{array} \right]^T\), \(\boldsymbol{v}= \left[ \begin{array}{lll} 1 & 1 & 1 \end{array} \right]^T\). Compute the 2-norms of \(\boldsymbol{u}, \boldsymbol{v}\). Determine the angle between \(\boldsymbol{u}, \boldsymbol{v}\).
Consider \(\boldsymbol{u}= \left[ \begin{array}{lll} 1 & 1 & 0 \end{array} \right]^T\), \(\boldsymbol{v}= \left[ \begin{array}{lll} 1 & 1 & 1 \end{array} \right]^T\). Define vector \(\boldsymbol{w}\) such that \(\boldsymbol{v}+\boldsymbol{w}\) is orthogonal to \(\boldsymbol{u}\). Write the equation to determine \(\boldsymbol{w}\), and then compute \(\boldsymbol{w}\).
Determine \(\boldsymbol{q}_1, \boldsymbol{q}_2\) to be of unit norm and in the direction of vectors \(\boldsymbol{u}, \boldsymbol{w}\) from Ex. 4. Form \(\hat{\boldsymbol{Q}} = \left[ \begin{array}{ll} \boldsymbol{q}_1 & \boldsymbol{q}_2 \end{array} \right]\). Compute \(\hat{\boldsymbol{Q}} \hat{\boldsymbol{Q}}^T\) and \(\widehat{\boldsymbol{Q} }^T \hat{\boldsymbol{Q}}\).
Determine vector \(\boldsymbol{q}_3\) orthonormal to vectors \(\boldsymbol{q}_1, \boldsymbol{q}_2\) from Ex. 5.
Establish whether vectors \(\boldsymbol{u}= \left[ \begin{array}{lll} 1 & 2 & 3 \end{array} \right]^T\), \(\boldsymbol{v}= \left[ \begin{array}{lll} - 3 & 1 & - 2 \end{array} \right]^T\), \(\boldsymbol{w}= \left[ \begin{array}{lll} 2 & - 3 & 1 \end{array} \right]^T\) all lie in the same plane within \(\mathbb{R}^3\).
Determine \(\boldsymbol{v}\) the reflection of vector \(\boldsymbol{u}= \left[ \begin{array}{ll} 1 & \sqrt{3} \end{array} \right]^T\) across vector \(\boldsymbol{w}= \left[ \begin{array}{ll} 1 & 1 \end{array} \right]^T\).
Determine \(\boldsymbol{w}\) the rotation of vector \(\boldsymbol{u}= \left[ \begin{array}{ll} 1 & \sqrt{3} \end{array} \right]^T\) by angle \(\theta = - \pi / 6\).
Compute \(\boldsymbol{z}=\boldsymbol{v}-\boldsymbol{w}\) with \(\boldsymbol{v}, \boldsymbol{w}\) from Ex. 8,9.
Find two linear combinations of vectors \(\boldsymbol{u}= \left[ \begin{array}{lll} 1 & 1 & 1 \end{array} \right]\), \(\boldsymbol{v}= \left[ \begin{array}{lll} 1 & 2 & 3 \end{array} \right]\) first with scaling coefficients \(\alpha = 2\), \(\beta = 1\), and then with scaling coefficients \(\alpha = 1\), \(\beta = 2\).
Express the above linear combinations \(\boldsymbol{B}\) as a matrix-matrix product \(\boldsymbol{B}=\boldsymbol{A}\boldsymbol{X}\). Define the column vectors of \(\boldsymbol{A}, \boldsymbol{X}\).
Consider \(\boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{m \times m}\). Which of the following matrices are always equal to \(\boldsymbol{C}= (\boldsymbol{A}-\boldsymbol{B})^2\)?
\(\boldsymbol{A}^2 -\boldsymbol{B}^2\)
\((\boldsymbol{B}-\boldsymbol{A})^2\)
\(\boldsymbol{A}^2 - 2\boldsymbol{A}\boldsymbol{B}+\boldsymbol{B}^2\)
\(\boldsymbol{A} (\boldsymbol{A}-\boldsymbol{B}) -\boldsymbol{B} (\boldsymbol{B}-\boldsymbol{A})\)
\(\boldsymbol{A}^2 -\boldsymbol{A}\boldsymbol{B}-\boldsymbol{B}\boldsymbol{A}+\boldsymbol{B}^2\)
Find the inverse of
Verify that the inverse of \(\boldsymbol{A}=\boldsymbol{I}-\boldsymbol{u}\boldsymbol{v}^T\) is
when \(\boldsymbol{v}^T \boldsymbol{u} \neq 1\).
Find \(\boldsymbol{A}^T, \boldsymbol{A}^{- 1}, (\boldsymbol{A}^{- 1})^T, (\boldsymbol{A}^T)^{- 1}\) for
Describe within \(\mathbb{R}^3\) the geometry of the column spaces of matrices
The vector subspaces of \(\mathbb{R}^2\) are lines, \(\mathbb{R}^2\) itself and \(Z = \left\{ \left[ \begin{array}{ll} 0 & 0 \end{array} \right]^T \right\}\). What are the vector subspaces of \(\mathbb{R}^3\)?
Reduce the following matrices to row echelon form
Determine the null space of