Vector Spaces

Prove that $\mathbb{R}^{n}$ under addition as defined in 1.2 is an abelian group.

Let

Is $W$ a vector space?

Prove 1.7.

Let $F$ be a field and $n$ be a non-negative integer. Prove that

is a vector space over $F$ using the definitions of polynomial addition and scalar multiplication.(Warning: be careful with the zero polynomial!)

Let the vector space

Let $S=\{x,x+2,x^{2}+2,x^{2}+2x+3\}$. Prove or disprove: $S$ is a spanning set for $V$.

For the vector space

over $\mathbb{Q}$, which of the following sets of vectors of $V$ is/are linearly independent?

1. $S=\{1,x+2\}$

2. $T=\{1,x+2,x^{2}+2x+3\}$

3. $U=\{1,x+2,x^{2}+2x+3,x^{2}-2x-3\}$

Prove that if the set $S$ is a basis for the vector space $V$ over a field $F$, then every vector in $V$ can be expressed uniquely as a linear combination of the vectors in $S$.

For the vector space $V$ given in 1.6, which of the following sets of vectors of $V$ is/are a basis for $V$?

1. $S=\{1,x+2\}$

2. $T=\{1,x+2,x^{2}+2x+3\}$

3. $U=\{1,x+2,x^{2}+2x+3,x^{2}-2x-3\}$

What is the dimension of the vector space $V$ in 1.6?

Let $U$ and $W$ be subspaces of a vector space $V$ over the field $F$.

1. Prove that $U\cap W$ is a subspace of $V$.

2. Define the set

   Prove that $U+W$ is a subspace of $V$.

Suppose $V$ is a vector space over the field $F$ with basis $\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}$. Let

Prove that $\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\}$ is also a basis for $V$.

Let $V=\mathbb{R}^{4}$ be a vector space over $\mathbb{R}$.

1. Let the set

   Show that $R$ is not a subspace of $V$.

2. Let the set

   Show that $S$ is a subspace of $V$. Find a basis for $S$ and hence state the value of $\dim(S)$.

Let $F$ be a field and $n$ be a positive integer. Define addition and scalar multiplication by

and

respectively, for any $(u_{1},u_{2},\dots,u_{n}),(v_{1},v_{2},\dots,v_{n})\in F^{n}$ and $\alpha\in F$. Prove that $F^{n}$ is a vector space over $F$ under the above definitions of addition and scalar multiplication.

The analogue of a group/ring homomorphism for vector spaces is called a linear transformation. If $V$ and $W$ are vector spaces over a field $F$, then the map $T:V\to W$ is a linear transformation if it preserves vector addition and scalar multiplication, that is,

for all $\mathbf{u},\mathbf{v}\in V$ and $\alpha\in F$.

To prove that two vector spaces are isomorphic, one needs to find a bijective linear transformation.

With this information, p…