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Section 7.5 Exercises
Exercise tags. The tags below identify the primary outcomes and competency types for each exercise group. Individual problems may also involve earlier outcomes.
Subsection Abstract vector spaces
Primary outcomes. U7-LO1, U7-LO2.
Competencies. P+J.
Subsection Subspaces, bases, and coordinates in nonstandard spaces
Primary outcomes. U7-LO1, U7-LO2.
Competencies. P+R+J.
Nicholson 6.2.2(d) math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.02%3A_Subspaces_and_Spanning_Sets/6.2E%3A_Subspaces_and_Spanning_Sets_Exercises
Nicholson 6.2.2(f) math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.02%3A_Subspaces_and_Spanning_Sets/6.2E%3A_Subspaces_and_Spanning_Sets_Exercises
Nicholson 6.3.2(d) math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.03%3A_Linear_Independence_and_Dimension/6.3E%3A_Linear_Independence_and_Dimension_Exercises
Subsection Inner product spaces
Primary outcomes. U7-LO3, U7-LO4.
Competencies. P+C+J.
Subsection Additional applied and computational problems
These exercises connect the main Unit 7 ideas: abstract vector spaces, matrix and polynomial examples, inner products, projection equations, Taylor and least-squares approximations, Gram-Schmidt in polynomial spaces, sampled polynomial regression, two-variable planes, Hessians, and code interpretation. Solutions are collected in Appendix
C.7 .
Checkpoint 7.5.1 . Matrix-space coordinates.
Let
\begin{equation*}
E_{11}=
\begin{bmatrix}
1\amp0\\
0\amp0
\end{bmatrix},
\quad
E_{12}=
\begin{bmatrix}
0\amp1\\
0\amp0
\end{bmatrix},
\quad
E_{21}=
\begin{bmatrix}
0\amp0\\
1\amp0
\end{bmatrix},
\quad
E_{22}=
\begin{bmatrix}
0\amp0\\
0\amp1
\end{bmatrix}.
\end{equation*}
Let
\begin{equation*}
A=
\begin{bmatrix}
2\amp-1\\
0\amp3
\end{bmatrix}.
\end{equation*}
Write
\(A\) as a linear combination of
\(E_{11},E_{12},E_{21},E_{22}\text{.}\)
Write the coordinate vector of
\(A\) in the ordered basis
\((E_{11},E_{12},E_{21},E_{22})\text{.}\)
What is
\(\dim M_{22}\text{?}\)
What is the zero vector in
\(M_{22}\text{?}\)
Tags. [U7-LO2 | P+R | Core]
Checkpoint 7.5.2 . Polynomial coordinates and subspaces.
Let
\(p(x)=2-3x+x^3\text{.}\)
Write the coordinate vector of
\(p\) in the ordered basis
\((1,x,x^2,x^3)\text{.}\)
Find a basis for
\(U=\{p\in\mathcal P_{\le3}:p(0)=0\}\text{.}\)
Find a basis for
\(W=\{p\in\mathcal P_{\le3}:p(0)=p'(0)=0\}\text{.}\)
Explain why
\(U\) and
\(W\) are subspaces.
Tags. [U7-LO1, U7-LO2 | P+R+J | Core]
Checkpoint 7.5.3 . Derivative and evaluation as linear maps.
Define
\begin{equation*}
D:\mathcal P_{\le3}\to\mathcal P_{\le2},\qquad D(p)=p',
\end{equation*}
and
\begin{equation*}
E_0:\mathcal P_{\le3}\to\mathbb R,\qquad E_0(p)=p(0).
\end{equation*}
Use the ordered bases \((1,x,x^2,x^3)\) for \(\mathcal P_{\le3}\) and \((1,x,x^2)\) for \(\mathcal P_{\le2}\text{.}\)
Explain why
\(D\) is linear.
Find the matrix of
\(D\) in these bases.
Explain why
\(E_0\) is linear.
Find a basis for the kernel of
\(E_0\text{.}\)
Tags. [U7-LO1, U7-LO2 | P+R+J | Core]
Checkpoint 7.5.4 . Matrix inner product.
Use the matrix inner product \(\langle A,B\rangle=\operatorname{tr}(A^TB)\text{.}\) Let
\begin{equation*}
A=
\begin{bmatrix}
1\amp2\\
0\amp-1
\end{bmatrix},
\qquad
B=
\begin{bmatrix}
3\amp0\\
4\amp1
\end{bmatrix}.
\end{equation*}
Compute
\(\langle A,B\rangle\text{.}\)
Compute
\(\|A\|\text{.}\)
Compute
\(d(A,B)\text{.}\)
Are
\(A\) and
\(B\) orthogonal?
Tags. [U7-LO3 | P+C | Core]
Checkpoint 7.5.5 . Same polynomial space, different inner products.
In
\(\mathcal P_{\le1}\text{,}\) let
\(p(x)=1+x\) and
\(q(x)=1-x\text{.}\)
Compute
\(\langle p,q\rangle_{\mathrm{coef}}\text{.}\)
Compute
\(\langle p,q\rangle_{L^2}\text{,}\) where
\(\langle p,q\rangle_{L^2}=\int_{-1}^{1}p(x)q(x)\,dx\text{.}\)
Are
\(p\) and
\(q\) orthogonal for the coefficient inner product?
Are
\(p\) and
\(q\) orthogonal for the
\(L^2[-1,1]\) inner product?
What does this show about the phrase “orthogonal polynomials”?
Tags. [U7-LO3 | C+P+R | Core]
Checkpoint 7.5.6 . Projection equations for the continuous least-squares line.
Let
\(p(x)=1+2x+x^2+x^3\text{,}\) and
\(U=\operatorname{span}\{1,x\}\text{.}\) Use the
\(L^2[-1,1]\) inner product.
Compute the Gram matrix for the ordered basis
\((1,x)\text{.}\)
Compute the right-hand side vector
\(\mathbf b\) for projecting
\(p\) onto
\(U\text{.}\)
Solve
\(G\mathbf c=\mathbf b\text{.}\)
What is the continuous least-squares line?
Tags. [U7-LO4, U7-LO5 | P+R | Core]
Checkpoint 7.5.7 . Taylor line versus fixed-interval least squares.
Find the Taylor line of
\(f\) at
\(0\text{.}\)
Find the
\(L^2[-1,1]\) -projection of
\(f\) onto
\(\mathcal P_{\le1}=\operatorname{span}\{1,x\}\text{.}\)
Explain why the answers differ.
Tags. [U7-LO5 | C+P | Core]
Checkpoint 7.5.8 . Taylor projection from a local inner product.
On \(\mathcal P_{\le3}\text{,}\) define
\begin{equation*}
\langle p,q\rangle_0
=
\sum_{k=0}^{3}
\frac{p^{(k)}(0)}{k!}
\frac{q^{(k)}(0)}{k!}.
\end{equation*}
Let \(p(x)=a_0+a_1x+a_2x^2+a_3x^3\text{.}\) Show that the orthogonal projection of \(p\) onto \(\mathcal P_{\le1}\) is \(a_0+a_1x\text{.}\) Then rewrite the answer using \(p(0)\) and \(p'(0)\text{.}\)
Tags. [U7-LO3, U7-LO5 | J+C | Core]
Checkpoint 7.5.9 . Shrinking least squares.
Let
\(P_t f\) denote the
\(L^2[-t,t]\) -projection of
\(f\) onto
\(\operatorname{span}\{1,x\}\text{.}\)
Show that
\(P_t(x^2)=\frac{t^2}{3}\text{.}\)
Show that
\(P_t(x^3)=\frac{3t^2}{5}x\text{.}\)
What happens to both projections as
\(t\to0\text{?}\)
How does this connect to Taylor approximation?
Tags. [U7-LO5 | P+C | Core]
Checkpoint 7.5.10 . Gram-Schmidt in \(\mathcal P_{\le3}\) .
Use the inner product
\(\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)\,dx\text{.}\) Apply Gram-Schmidt to
\(1,\ x,\ x^2,\ x^3\) through the
\(x^3\) step.
Explain why
\(1\) and
\(x\) are orthogonal.
Derive
\(x^2-\frac13\text{.}\)
Derive
\(x^3-\frac35x\text{.}\)
Why does an orthogonal basis make projection easier?
Tags. [U7-LO6 | P+C | Core]
Checkpoint 7.5.11 . Code interpretation: polynomial regression.
A = np.column_stack([np.ones_like(xs), xs, xs**2])
c = np.linalg.lstsq(A, y, rcond=None)[0]
r = y - A @ c
A.T @ r
What polynomial subspace is represented by the columns of
A?
What does the vector
c store?
Why does
\(A^T\mathbf r\approx\mathbf0\) not mean
\(\mathbf r\approx\mathbf0\text{?}\)
Tags. [U7-LO4, U7-LO5 | C+T | Core]
Checkpoint 7.5.12 . Two-variable tangent plane and sampled design matrix.
Let
\(f(x,y)=1+2x-y+x^2+xy+2y^2+x^3\text{.}\)
Compute the tangent plane to
\(f\) at
\((0,0)\text{.}\)
Write the residual
\(r_T=f-T\text{.}\)
What conditions do
\(r_T(0,0)\) and
\(\nabla r_T(0,0)\) satisfy?
For sampled points
\((x_i,y_i)\text{,}\) write the design matrix for fitting a plane
\(q(x,y)=\alpha+\beta x+\gamma y\text{.}\)
Write the sampled residual-orthogonality equations.
Tags. [U7-LO5 | C+R+P | Core]
Checkpoint 7.5.13 . Hessian as a quadratic form.
