Definition and Examples

Decide whether each subset of $\mathbb{R}^{3}$ is linearly dependent or linearly independent.

• (a)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ -3\\ 5\end{pmatrix},\begin{pmatrix}2\\ 2\\ 4\end{pmatrix},\begin{pmatrix}4\\ -4\\ 14\end{pmatrix}\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 7\\ 7\end{pmatrix},\begin{pmatrix}2\\ 7\\ 7\end{pmatrix},\begin{pmatrix}3\\ 7\\ 7\end{pmatrix}\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt\begin{pmatrix}0\\ 0\\ -1\end{pmatrix},\begin{pmatrix}1\\ 0\\ 4\end{pmatrix}\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt\begin{pmatrix}9\\ 9\\ 0\end{pmatrix},\begin{pmatrix}2\\ 0\\ 1\end{pmatrix},\begin{pmatrix}3\\ 5\\ -4\end{pmatrix},\begin{pmatrix}12\\ 12\\ -1\end{pmatrix}\hskip 1.00006pt\}$

Which of these subsets of $\mathcal{P}_{3}$ are linearly dependent and which are independent?

• $\{\hskip 1.00006pt3-x+9x^{2},5-6x+3x^{2},1+1x-5x^{2}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt-x^{2},1+4x^{2}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt2+x+7x^{2},3-x+2x^{2},4-3x^{2}\hskip 1.00006pt\}$

• $\{\hskip 1.00006pt8+3x+3x^{2},x+2x^{2},2+2x+2x^{2},8-2x+5x^{2}\hskip 1.00006pt\}$

Determine if each set is linearly independent in the natural space.

• (a)  $\{\hskip 1.00006pt\begin{pmatrix}1\\ 2\\ 0\end{pmatrix},\begin{pmatrix}-1\\ 1\\ 0\end{pmatrix}\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt(\begin{matrix}1&3&1\end{matrix}),(\begin{matrix}-1&4&3\end{matrix}),(\begin{matrix}-1&11&7\end{matrix})\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006pt\begin{pmatrix}5&4\\ 1&2\end{pmatrix},\begin{pmatrix}0&0\\ 0&0\end{pmatrix},\begin{pmatrix}1&0\\ -1&4\end{pmatrix}\hskip 1.00006pt\}$

Prove that each set $\{\hskip 1.00006ptf,g\hskip 1.00006pt\}$ is linearly independent in the vector space of all functions from $\mathbb{R}^{+}$ to $\mathbb{R}$.

• $f(x)=x$ and $g(x)=1/x$

• $f(x)=\cos(x)$ and $g(x)=\sin(x)$

• $f(x)=e^{x}$ and $g(x)=\ln(x)$

Which of these subsets of the space of real-valued functions of one real variable is linearly dependent and which is linearly independent? (We have abbreviated some constant functions; e.g., in the first item, the ‘$2$’ stands for the constant function $f(x)=2$.)

• (a)  $\{\hskip 1.00006pt2,4\sin^{2}(x),\cos^{2}(x)\hskip 1.00006pt\}$     (b)  $\{\hskip 1.00006pt1,\sin(x),\sin(2x)\hskip 1.00006pt\}$     (c)  $\{\hskip 1.00006ptx,\cos(x)\hskip 1.00006pt\}$     (d)  $\{\hskip 1.00006pt(1+x)^{2},x^{2}+2x,3\hskip 1.00006pt\}$     (e)  $\{\hskip 1.00006pt\cos(2x),\sin^{2}(x),\cos^{2}(x)\hskip 1.00006pt\}$     (f)  $\{\hskip 1.00006pt0,x,x^{2}\hskip 1.00006pt\}$

Does the equation $\sin^{2}(x)/\cos^{2}(x)=\tan^{2}(x)$ show that this set of functions $\{\hskip 1.00006pt\sin^{2}(x),\cos^{2}(x),\tan^{2}(x)\hskip 1.00006pt\}$ is a linearly dependent subset of the set of all real-valued functions with domain the interval $(-\pi/2..\pi/2)$ of real numbers between $-\pi/2$ and $\pi/2)$?

Is the $xy$-plane subset of the vector space $\mathbb{R}^{3}$ linearly independent?

Show that the nonzero rows of an echelon form matrix form a linearly independent set.

• Show that if the set $\{\hskip 1.00006pt\vec{u},\vec{v},\vec{w}\hskip 1.00006pt\}$ is linearly independent then so is the set $\{\hskip 1.00006pt\vec{u},\vec{u}+\vec{v},\vec{u}+\vec{v}+\vec{w}\hskip 1.00006pt\}$.

• What is the relationship between the linear independence or dependence of $\{\hskip 1.00006pt\vec{u},\vec{v},\vec{w}\hskip 1.00006pt\}$ and the independence or dependence of $\{\hskip 1.00006pt\vec{u}-\vec{v},\vec{v}-\vec{w},\vec{w}-\vec{u}\hskip 1.00006pt\}$?

Example 1.11 shows that the empty set is linearly independent.

• When is a one-element set linearly independent?

• How about a set with two elements?

In any vector space $V$, the empty set is linearly independent. What about all of $V$?

Show that if $\{\hskip 1.00006pt\vec{x},\vec{y},\vec{z}\hskip 1.00006pt\}$ is linearly independent then so are all of its proper subsets: $\{\hskip 1.00006pt\vec{x},\vec{y}\hskip 1.00006pt\}$, $\{\hskip 1.00006pt\vec{x},\vec{z}\hskip 1.00006pt\}$, $\{\hskip 1.00006pt\vec{y},\vec{z}\hskip 1.00006pt\}$, $\{\hskip 1.00006pt\vec{x}\hskip 1.00006pt\}$,$\{\hskip 1.00006pt\vec{y}\hskip 1.00006pt\}$, $\{\hskip 1.00006pt\vec{z}\hskip 1.00006pt\}$, and $\{\hskip 1.00006pt\hskip 1.00006pt\}$. Is that ‘only if’ also?

• Show that this

  is a linearly independent subset of $\mathbb{R}^{3}$.

• Show that

  is in the span of $S$ by finding $c_{1}$ and $c_{2}$ giving a linear relationship.

  Show that the pair $c_{1},c_{2}$ is unique.

• Assume that $S$ is a subset of a vector space and that $\vec{v}$ is in $[S]$, so that $\vec{v}$ is a linear combination of vectors from $S$. Prove that if $S$ is linearly independent then a linear combination of vectors from $S$ adding to $\vec{v}$ is unique (th…

Prove that a polynomial gives rise to the zero function if and only if it is the zero polynomial. (Comment. This question is not a Linear Algebra matter but we often use the result. A polynomial gives rise to a function in the natural way: $x\mapsto c_{n}x^{n}+\dots+c_{1}x+c_{0}$.)

Return to Section 1.2 and redefine point, line, plane, and other linear surfaces to avoid degenerate cases.

• Show that any set of four vectors in $\mathbb{R}^{2}$ is linearly dependent.

• Is this true for any set of five? Any set of three?

• What is the most number of elements that a linearly independent subset of $\mathbb{R}^{2}$ can have?

Is there a set of four vectors in $\mathbb{R}^{3}$ such that any three form a linearly independent set?

Must every linearly dependent set have a subset that is dependent and a subset that is independent?

In $\mathbb{R}^{4}$ what is the biggest linearly independent set you can find? The smallest? The biggest linearly dependent set? The smallest? (‘Biggest’ and ‘smallest’ mean that there are no supersets or subsets with the same property.)

Linear independence and linear dependence are properties of sets. We can thus naturally ask how the properties of linear independence and dependence act with respect to the familiar elementary set relations and operations. In this body of this subsection we have covered the subset and superset relations. We can also consider the operations of intersection, complementation, and union.

• How does li…

Continued from prior exercise. What is the interaction between the property of linear independence and the operation of union?

• We might conjecture that the union $S\cup T$ of linearly independent sets is linearly independent if and only if their spans have a trivial intersection $[S]\cap[T]=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}$. What is wrong with this argument for the ‘if’ direction of that conjecture? “If the union $S\cup T$ is linearly independent then th…

For Corollary 1.17,

• fill in the induction for the proof;

• give an alternate proof that starts with the empty set and builds a sequence of linearly independent subsets of the given finite set until one appears with the same span as the given set.

With a some calculation we can get formulas to determine whether or not a set of vectors is linearly independent.

• Show that this subset of $\mathbb{R}^{2}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ c\end{pmatrix},\begin{pmatrix}b\\ d\end{pmatrix}\hskip 1.00006pt\}$

  is linearly independent if and only if $ad-bc\neq 0$.

• Show that this subset of $\mathbb{R}^{3}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ d\\ g\end{pmatrix},\begin{pmatrix}b\\ e\\ h\end{pmatrix},\begin{pmatrix}c\\ f\\ i\end{pmatrix}\hskip 1.00006pt\}$

  is linearly independent iff $aei+bfg+cdh-hfa-idb-gec\neq 0$.

• When is this subset of $\mathbb{R}^{3}$

  $\{\hskip 1.00006pt\begin{pmatrix}a\\ d\\ g\end{pmatrix},\begin{pmatrix}b\\ e\\ h\end{pmatrix}\hskip 1.00006pt\}$

  linearly independent?

• This is an opinion question: for a set of four vectors from $\mathbb{R}^{4}$, must there be a formula involving t…

• Prove that a set of two perpendicular nonzero vectors from $\mathbb{R}^{n}$ is linearly independent when $n>1$.

• What if $n=1$? $n=0$?

• Generalize to more than two vectors.

Consider the set of functions from the interval $(\,-1\ldots 1\,)\subseteq\mathbb{R}$ to $\mathbb{R}$.

• Show that this set is a vector space under the usual operations.

• Recall the formula for the sum of an infinite geometric series: $1+x+x^{2}+\cdots=1/(1-x)$ for all $x\in(-1..1)$. Why does this not express a dependence inside of the set $\{\hskip 1.00006ptg(x)=1/(1-x),f_{0}(x)=1,f_{1}(x)=x,f_{2}(x)=x^{2},\ldots\hskip 1.00006pt\}$ (in the vector space that we are considering)? (Hint. Review the definition of linear combination.)

• Show that the set in the prior item i…

Show that, where $S$ is a subspace of $V$, if a subset $T$ of $S$ is linearly independent in $S$ then $T$ is also linearly independent in $V$. Is that ‘only if’?