Problems

01

Unique coefficients on independent vectors

Suppose the vectors $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_k\}$ are independent. Show that the coefficients on these vectors are unique, in any linear combination. This means, for example, that if we know

$$x_1{\mathbf{v}}_1+\cdots +x_k{\mathbf{v}}_k=\mathbf{w}=y_1{\mathbf{v}}_1+\cdots +y_k{\mathbf{v}}_k,$$

then in fact we have $x_i=y_i$ for all $i$. In other words, there is at most one way to write a vector as a linear combination of some independent vectors.

02

Neutralizing a matrix row

Suppose that the row vectors of a matrix $M$ are dependent. Explain precisely how (in terms of a given dependency relation) you can transform $M$ into a matrix having a row of all zeros by performing some combination of row-scale and row-add operations to the rows of $M$.

03

Column dependence and multiple solutions

Suppose that the columns of a matrix $A$ are dependent. Consider equations of the form $A\mathbf{x}=\mathbf{b}$ for $\mathbf{x}$ a variable vector, and $\mathbf{b}$ a given constant vector. Show that if there is a solution $\mathbf{x}$ to this equation, then there must be more than one solution $\mathbf{x}$. (Because of the dependency of the columns of $A$.)

(Hint: because the columns of $A$ are dependent, (explain how) you can find a specific vector ${\mathbf{x}}^{\prime }\ne 0$ with the property that $A{\mathbf{x}}^{\prime }=0$. Combine ${\mathbf{x}}^{\prime }$ with a given solution to the equation to produce a second solution. Can you always get an infinite set of solutions?)

04

Checking independence

Is the following set of vectors linearly independent?

$$\left\{\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right),\left(\begin{array}{c}4\\ 5\\ 6\end{array}\right),\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right)\right\}$$

If it’s dependent, find a dependency relation.

(You are encouraged to do this problem without Steinitz Exchange and dimensional reasoning: simply solve a system of equations instead.)

05

Vector operations according to components in another basis

Show that the list of components of a vector in some basis $\mathcal{B}$ are added and scaled componentwise.

This means: given two vectors, if you express them in the basis $\mathcal{B}$, and add the components in this basis, you will get the same result as if you first added the two vectors (according to the definition of adding components within ${\mathbb{R}}^n$) and then expressed the result in the basis $\mathcal{B}$. And similarly for scaling.