Math 260, Fall 2018, Assignment 7
From cartan.math.umb.edu
Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.
- - Lord Kelvin
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Linear subspace (of $\mathbb{R}^n$).
- Linear combination (of vectors $\vec{v}_1,\dots,\vec{v}_k\in\mathbb{R}^n$).
- Span (of a set of vectors $\vec{v}_1,\dots,\vec{v}_k$).
- Image (of a linear transformation).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem relating spans to linear subspaces.
- Theorem relating images to linear subspaces.
Solve the following problems:[edit]
- Section 3.2, problems 1, 2, 3, 5, and 6.
- In class, we described all linear subspaces of $\mathbb{R}^2$ and $\mathbb{R}^3$. Use similar reasoning to describe all linear subspaces of $\mathbb{R}^1$.
- Determine whether the vector $\vec{v}=\begin{bmatrix}4\\3\\5\end{bmatrix}$ lies in the plane spanned by $\vec{v}_1=\begin{bmatrix}1\\0\\2\end{bmatrix}$ and $\vec{v}_2=\begin{bmatrix}2\\3\\1\end{bmatrix}$. (Hint: you need to determine whether there is any choice of scalars $c_1,c_2$ making $\vec{v}=c_1\vec{v}_1+c_2\vec{v}_2$. This condition is equivalent to a system of linear equations in the unknowns $c_1,c_2$. How can you tell whether such a system has any solutions?)
- Determine whether the vector $\vec{v}=\begin{bmatrix}4\\3\\6\end{bmatrix}$ lies in the plane spanned by $\vec{v}_1=\begin{bmatrix}1\\0\\2\end{bmatrix}$ and $\vec{v}_2=\begin{bmatrix}2\\3\\1\end{bmatrix}$.
