MATH347DS

MATH347.MM.23 Final Examination

Final examination allows for makeup of deducted homework points, HW points = 72, Exam points = 40.

1Basic concepts

Answer True/False or Yes/No with a brief explanation of your answer.

Question 1. The polynomial $p (t) = t^{2} - t + 1$ is a linear combination of the basis functions ${1, t, t^{2}, t^{3},}$ .

Answer 1.

Question 2. Let $x, y, z \in ℝ^{2}$ . Does $x^{T} y = x^{T} z$ imply that $y = z ?$

Answer 2.

Question 3. If S spans the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way.

Answer 3.

Question 4. In some vector space $a, b$ are scalars, $𝒖$ is a vector. Does $a 𝒖 = b 𝒖$ imply $a = b$ ?

Answer 4.

Question 5. In some vector space $a$ is a scalar, $𝒖, 𝒗$ are vectors. Does $a 𝒖 = a 𝒗$ imply $𝒖 = 𝒗$ ?

Answer 5.

Question 6. The cosine of the angle between vectors u,v is the scalar product u^Tv.

Answer 6.

Question 7. The singular values of matrix A^T are the same as those of A.

Answer 7.

Question 8. For any vector u we can find a unit vector in the same direction.

Answer 8.

Question 9. The singular values of A² are the squares of the singular values of A.

Answer 9.

Question 10. The eigenvalues of A² are the squares of the eigenvalues of A.

Answer 10.

Primer on factorizations of matrix $𝑨 \in ℝ^{m \times n}$

\begin{array}{llll} 𝑸 𝑹 = 𝑨 & 𝑳 𝑼 = 𝑨 & 𝑨 𝑿 = 𝑿 𝚲 & 𝑨 = 𝑼 𝚺 𝑽^{T} \\ 𝑸 \in ℝ^{m \times n} & 𝑳 \in ℝ^{m \times n} & m = n & 𝑼 \in ℝ^{m \times m} \\ 𝑸^{T} 𝑸 = 𝑰 & 𝑼 \in ℝ^{n \times n} & 𝑿 \in ℂ^{m \times m} & 𝑽 \in ℝ^{n \times n} \\ 𝑹 \in ℝ^{n \times n} & 𝑳 lower triangular & 𝚲 = diag (λ_{1}, . ., λ_{m}) & 𝑼^{T} 𝑼 = 𝑰_{m}, 𝑽^{T} 𝑽 = 𝑰_{n} \\ 𝑹 upper triangular & 𝑼 upper triangular & λ_{j} \in ℂ & 𝚺 = diag (σ_{1}, σ_{2}, . .) \in ℝ_{+}^{m \times n} \\ 𝑿 nonsingular \Rightarrow 𝑿 𝚲 𝑿^{- 1} = 𝑨 & σ_{1} ⩾ σ_{2} ⩾ \dots ⩾ σ_{r} > 0 \end{array}

Find the singular value decomposition of $𝑨 = [\begin{array}{lll} 1 & 1 & 1 \end{array}] \in ℝ^{1 \times 3}$ .
Find the eigendecomposition of $𝑨^{T} 𝑨$ , $𝑨 = [\begin{array}{lll} 1 & 1 & 1 \end{array}] \in ℝ^{1 \times 3}$ .
Find the projection of
$𝒗 \in [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] \in ℝ^{3}$
onto $C (𝑨)$ with
$𝑨 = [\begin{array}{ll} 1 & - 1 \\ 0 & 2 \\ - 1 & - 1 \end{array}] .$