Study Notes on Vector Spaces and Linear Transformations

Definition of Vector Spaces

• A nonempty set $V$ is said to be a vector space over a field $K$ if it satisfies the following axioms:

  • $(V, +)$ is an abelian group.

  • Scalar Multiplication Closure: $k \cdot u \in V$ for all $u \in V$ and $k \in F$.

  • Distributive Law 1: $k \cdot (u + v) = k \cdot u + k \cdot v$.

  • Distributive Law 2: $(k_1 + k_2) \cdot u = k_1 \cdot u + k_2 \cdot u$.

  • Associativity of Scalars: $(k_1 k_2) \cdot u = k_1 \cdot (k_2 \cdot u)$.

  • Scalar Identity: $1 \cdot u = u$.

Subspaces

• Definition: A non-empty subset of a vector space that satisfies all the properties of the vector space over the same field under the same vector addition and scalar multiplication is called a subspace.

• Property-1: A nonempty set $W$ is a subspace of a vector space $V$ over a field $K$ if:

  • The zero vector is in $W$.

  • Closure under addition and scalar multiplication: $u + v \in W$ and $ku \in W$ for all $u, v \in W$ and $k \in K$.

  • Equivalent condition: $au + bv \in W$ for $u, v \in W$ and $a, b \in K$.

• Problems on Subspaces:

  • Determine if the following subsets of $\mathbb{R}^2$ are subspaces:

    • $W_1$: Vectors of the form $(x, y)$, where $x \ge 0$.

    • $W_2$: Vectors of the form $(x, y)$, where $x \ge 0$ and $y \ge 0$.

    • $W_3$: Vectors of the form $(x, y)$, where $x = 0$.

    • $W_4$: Vectors of the form $(x, y)$, where $xy = 0$.

    • $W_5$: Vectors of the form $(3s, 2 + 5s)$ for $s \in \mathbb{R}$.

  • Determine if $W$ is a subspace of $\mathbb{R}^3$ consisting of vectors $(a, b, c)$ such that:

    • $a = 3b$

    • $a \le b \le c$

    • $ab = 0$

    • $a + b + c = 0$

    • $b = a^2$

    • $a = 2b = 3c$

    • $a + b + c = 3$

    • $a \ge 0$

    • $a^2 + b^2 + c^2 \le 1$

  • Determine if $W = \{(\alpha, 4\alpha, 5\alpha) : \alpha \text{ is any scalar}\}$ is a subspace of $\mathbb{R}^3$.

  • Determine if the subset of matrices of the form $\begin{bmatrix} a & a \\ b & b \end{bmatrix}$ is a subspace.

  • Determine if $W = \{p(x) = a + bx^2 \mid a, b \in \mathbb{R}\}$ is a subspace of $P_2$.

Linear Combination and Linear Span

• Linear Combination: A vector $v$ is said to be the linear combination of a set of vectors $S = \{u_1, u_2, \dots, u_n\}$ if $v = c_1 u_1 + c_2 u_2 + \dots + c_n u_n$ for all $c_i$ in $F$.

• Spanning Sets or Linear Span: Consider a subset $S = \{u_1, u_2, \dots, u_n\}$ of a vector space $V$ over a field $F$. The collection of all possible linear combinations of vectors in $S$ is called the linear span of $S$, denoted by $span(S) = span(u_i)$. $span(S)$ is a subspace of $V$.

• Problems on Spanning Sets:

  • Verify if $v$ in $\mathbb{R}^3$ is a linear combination of vectors $u_1, u_2,$ and $u_3$ (or in $Span(S)$):

    • a. $v = (1, -2, 5), u_1 = (1, 1, 1), u_2 = (1, 2, 3), u_3 = (2, -1, 1)$.

    • b. $v = (2, -5, 3), u_1 = (1, -3, 2), u_2 = (2, -4, -1), u_3 = (1, -5, 7)$.

    • c. $v = (4, -9, 2), u_1 = (1, 2, -1), u_2 = (1, 4, 2), u_3 = (1, -3, 2)$.

    • d. $v = (1, 3, 2), u_1 = (1, 2, 1), u_2 = (2, 6, 5), u_3 = (1, 7, 8)$.

    • e. $v = (1, 4, 6), u_1 = (1, 1, 2), u_2 = (2, 3, 5), u_3 = (3, 5, 8)$.

  • Find $k$ such that $v = (1, -2, k)$ is a linear combination of $(3, 0, -2)$ and $(2, -1, -5)$.

  • Express polynomial $v = t^2 + 4t - 3$ in $P(t)$ as a linear combination of $p_1 = t^2 - 2t + 5$, $p_2 = 2t^2 - 3t$, and $p_3 = t + 1$.

  • Express matrix $M = \begin{bmatrix} 4 & 7 \\ 7 & 9 \end{bmatrix}$ as a linear combination of $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, and $C = \begin{bmatrix} 1 & 1 \\ 4 & 5 \end{bmatrix}$.

  • Check if the following polynomials are in $span(p_1, p_2, p_3)$ where $p_1 = t^2 + t^2$, $p_2 = t^2 + t$, and $p_3 = t + t$:

    • i) $p(t) = t^2 + t^2 + t$

    • ii) $p(t) = 2t^2 - t$

    • iii) $p(t) = 2t^2 + 1$

Linear Dependence and Independence

• Definition: Consider $S = \{u_1, u_2, \dots, u_n\}$ as a subset of a vector space $V$.

  • Linearly Dependent: The set $S$ is linearly dependent if there exists at least one non-zero scalar $c_i$ such that $c_1 u_1 + c_2 u_2 + \dots + c_n u_n = 0$.

  • Linearly Independent: If such non-zero scalars do not exist (all $c_i = 0$), the set is linearly independent.

• Key Properties:

  • Property-1: Any set of vectors containing the zero vector is a linearly dependent set.

  • Property-2: Any set with more than one vector is linearly dependent if and only if one of them is a linear combination of the other vectors.

• Problems on Linear Dependence:

  • Determine linear dependence in $\mathbb{R}^3$:

    • a) $u_1 = (1, 2, 3), u_2 = (0, 0, 0), u_3 = (1, 5, 6)$.

    • b) $u = (1, 1, 2), v = (2, 3, 1), w = (4, 5, 5)$.

    • c) $u_1 = (1, 2, 5), u_2 = (1, 3, 1), u_3 = (2, 5, 7), u_4 = (3, 1, 4)$.

    • d) $u_1 = (1, 2, 5), u_2 = (2, 5, 1), u_3 = (1, 5, 2)$.

  • Determine linear dependence in $\mathbb{R}^4$:

    • a. $(1, 2, -3, 1), (3, 7, 1, -2), (1, 3, 7, -4)$.

    • b. $(1, 3, 1, -2), (2, 5, -1, 3), (1, 3, 7, -2)$.

  • Determine linear dependence for polynomials in $P(t)$:

    • a. $u = t^3 - 4t^2 + 3t + 3, v = t^3 + 2t^2 + 4t - 1, w = 2t^3 - t^2 - 3t + 5$.

    • b. $u = t^3 - 5t^2 - 2t + 3, v = t^3 - 4t^2 - 3t + 4, w = 2t^3 - 17t^2 - 7t + 9$.

    • c. $u = t^2 + t, v = 2t^2 + 2t, w = t^2 + t$.

    • d. $u = 1, v = t, w = 1 + t$.

Basis and Dimension

• Basis Definition: A nonempty subset $B = \{u_1, u_2, \dots, u_n\}$ of a vector space $V$ over a field $K$ is a basis of $V$ if:

  • $B$ is a linearly independent set.

  • $span(B) = V$.

• Dimension Definition: The number of vectors in the basis is called the dimension of $V$, denoted by $dim(V) = n$.

• Key Properties:

  • Property-1: Any basis of a vector space will have the same number of vectors.

  • Property-2: In an $n$-dimensional vector space, any set containing more than $n$ vectors is linearly dependent.

  • Property-3: Any linearly independent set with less than $n$ vectors cannot span the vector space $V$, but can be extended to a basis.

• Problems on Basis/Dimension:

  • Check if these form a basis of $\mathbb{R}^3$:

    • a. $(1, 1, 1), (1, 0, 1)$.

    • b. $(1, 1, 1), (1, 2, 3), (2, -1, 1)$.

  • Check if $(1, 1, 1, 1), (1, 2, 3, 2), (2, 5, 6, 4), (2, 6, 8, 5)$ form a basis of $\mathbb{R}^4$. If not, find the dimension of their span.

  • Find basis and dimension for row space, column space, and null space of matrices:

    • Hint: $rowsp(A) = span(R_i)$, $colsp(A) = span(C_i)$, $nullsp(A) = \text{solution space of } AX = 0$.

    • i. $\begin{bmatrix} 1 & 2 & -1 & -2 & 0 \\ 2 & 4 & -1 & 1 & 0 \\ 3 & 6 & -1 & 4 & 0 \\ 0 & 0 & 1 & 5 & 0 \end{bmatrix}$

    • ii. $\begin{bmatrix} 1 & -3 & 4 & -2 & 5 & 4 \\ 2 & -6 & 9 & -1 & 8 & 2 \\ 2 & -6 & 9 & 1 & 9 & 7 \\ -1 & 3 & -4 & 2 & -5 & -4 \end{bmatrix}$

    • iii. $\begin{bmatrix} 1 & -1 & 1 & 3 & 2 \\ 2 & -1 & 1 & 5 & 1 \\ 0 & 1 & -1 & -1 & -3 \end{bmatrix}$

    • iv. $\begin{bmatrix} 0 & 0 & 3 & 1 & 4 \\ 1 & 3 & 1 & 2 & 1 \\ 3 & 9 & 4 & 5 & 2 \\ 4 & 12 & 8 & 8 & 7 \end{bmatrix}$

  • Extension: Extend $\{u_1 = (1, 1, 1, 1), u_2 = (2, 2, 3, 4)\}$ to a basis of $\mathbb{R}^4$.

  • Find a subset of $S$ that forms a basis for $W = span(u_i)$:

    • a. $u_1 = (1, 1, 1, 2, 3), u_2 = (1, 2, -1, -2, 1), u_3 = (3, 5, -1, -2, 5), u_4 = (1, 2, 1, -1, 4)$.

    • b. $u_1 = (1, -2, 1, 3, -1), u_2 = (-2, 4, -2, -6, 2), u_3 = (1, -3, 1, 2, 1), u_4 = (3, -7, 3, 8, -1)$.

  • Find basis and dimension of subspace $W$ spanned by:

    • a. $u = t^3 + 2t^2 - 2t + 1, v = t^3 + 3t^2 - 3t + 4, w = 2t^3 + t^2 - 7t - 7$ in $P_3(t)$.

    • b. $u = t^3 + t^2 - 3t + 2, v = 2t^3 + t^2 + t - 4, w = 4t^3 + 3t^2 - 5t + 2$ in $P_3(t)$.

    • c. $A = \begin{bmatrix} 1 & -5 \\ -4 & 2 \end{bmatrix}, B = \begin{bmatrix} 1 & 1 \\ -1 & 5 \end{bmatrix}, C = \begin{bmatrix} 2 & -4 \\ -5 & 7 \end{bmatrix}, D = \begin{bmatrix} 1 & -7 \\ -5 & 1 \end{bmatrix}$ in $V = M_{2,2}$.

Coordinate Vectors

• Definition: Let $V$ be an $n$-dimensional vector space over $K$ with basis $S = \{u_1, u_2, \dots, u_n\}$. Any vector $v \in V$ can be expressed as $v = c_1 u_1 + c_2 u_2 + \dots + c_n u_n$. The scalars $c_1, c_2, \dots, c_n$ are coordinates of $v$ relative to basis $S$. The coordinate vector is denoted $[v]_S = [c_1, c_2, \dots, c_n]$.

• Problems on Coordinate Vectors:

  • Given basis $S = \{u_1 = (1, 2), u_2 = (4, 7)\}$ of $\mathbb{R}^2$, find $[v]_S$ for $v = (5, 3)$.

  • Find coordinate vector of $v = (5, 3, 4)$ in $\mathbb{R}^3$ relative to:

    • i) $S = \{u_1 = (1, 1, 0), u_2 = (1, 1, 0), u_3 = (0, 1, 1)\}$.

    • ii) $S = \{u_1 = (1, 1, 1), u_2 = (1, 1, 0), u_3 = (1, 0, 0)\}$.

  • Find $[v]_S$ if $v = (1, 2, 5)$ and $S = \{u_1 = (1, 1, 1), u_2 = (1, 2, 3), u_3 = (2, -1, 1)\}$.

  • Find $[v]_S$ if $v = (2, 7, 4)$ and $S = \{u_1 = (1, 2, 0), u_2 = (1, 3, 2), u_3 = (0, 1, 3)\}$.

  • Coordinates of arbitrary $v = (10, 4, 2, 0)$ in $\mathbb{R}^4$ relative to $S = \{(1, 1, 1, 1), (1, 1, -1, -1), (1, -1, 1, -1), (1, -1, -1, 1)\}$.

  • Coordinates of arbitrary $v = (4, 7, 7, 9)$ in $\mathbb{R}^4$ relative to $S = \{(1, 1, 1, 1), (1, 2, 3, 4), (1, 1, 4, 5), (0, 0, 0, 1)\}$.

  • For basis $S = \{(t - 1)^3, (t - 1)^2, (t - 1), 1\}$ of $P_3(t)$, find $[v]_S$ for $v = t^3 + 3t^2 - 4t + 25$.

  • For basis $S = \{t^2 + 1, t - 1, (t - 1)^2\}$ of $P_2(t)$, find $[v]_S$ for $v = 2t^2 + 5t + 9$.

  • For basis $S = \{t^3 + t^2, t^2 + t, t + 1, 1\}$ of $P_3(t)$, find $[v]_S$ for $v = t^3 + 3t^2 - 4t + 25$.

  • Find coordinate vector for $A = \begin{bmatrix} 2 & 3 \\ 4 & -7 \end{bmatrix}$ in $M_{2,2}$ relative to:

    • i) Basis $S = \left\{ \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \right\}$.

    • ii) Usual basis $S = \left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}$.

  • For $M = M_{2,2}$, find $[A]_S$ for $A = \begin{bmatrix} 2 & 3 \\ 4 & -7 \end{bmatrix}$ relative to $S = \left\{ \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \right\}$.