Linear Algebra - Vector Spaces & Transformations Flashcards

Question 1

Define a Vector Space.

Accepted Answer

A set $V$ (whose elements are called vectors) over a field $F$ (whose elements are called scalars) with two operations: vector addition and scalar multiplication, satisfying 10 axioms.

Question 2

List the closure axioms for a vector space.

Accepted Answer

1. Closure under addition: For all $\mathbf{u}, \mathbf{v} \in V$, $\mathbf{u} + \mathbf{v} \in V$. 
2. Closure under scalar multiplication: For all $\mathbf{u} \in V$ and scalar $c \in F$, $c\mathbf{u} \in V$.

Question 3

List the commutative and associative axioms for vector addition.

Accepted Answer

3. Commutativity: For all $\mathbf{u}, \mathbf{v} \in V$, $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$. 
4. Associativity: For all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$, $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.

Question 4

List the identity and inverse axioms for vector addition.

Accepted Answer

5. Zero vector: There exists a zero vector $\mathbf{0} \in V$ such that for all $\mathbf{u} \in V$, $\mathbf{u} + \mathbf{0} = \mathbf{u}$. 
6. Additive inverse: For each $\mathbf{u} \in V$, there exists an additive inverse $-\mathbf{u} \in V$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$.

Question 5

List the distributive axioms for scalar multiplication.

Accepted Answer

7. Distributivity over vector addition: For all $c \in F$ and $\mathbf{u}, \mathbf{v} \in V$, $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$. 
8. Distributivity over scalar addition: For all $c, d \in F$ and $\mathbf{u} \in V$, $(c+d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$.

Question 6

List the associative and identity axioms for scalar multiplication.

Accepted Answer

9. Associativity: For all $c, d \in F$ and $\mathbf{u} \in V$, $c(d\mathbf{u}) = (cd)\mathbf{u}$. 
10. Multiplicative identity: For the multiplicative identity $1 \in F$ and all $\mathbf{u} \in V$, $1\mathbf{u} = \mathbf{u}$.

Question 7

What is a Subspace?

Accepted Answer

A subset $W$ of a vector space $V$ that is itself a vector space under the operations defined on $V$. It must contain the zero vector and be closed under vector addition and scalar multiplication.

Question 8

State the three criteria for a subset $W$ to be a Subspace of $V$.

Accepted Answer

1. $W$ is non-empty (contains the zero vector, $\mathbf{0} \in W$). 
2. $W$ is closed under vector addition: For all $\mathbf{u}, \mathbf{v} \in W$, $\mathbf{u} + \mathbf{v} \in W$. 
3. $W$ is closed under scalar multiplication: For all $\mathbf{u} \in W$ and scalar $c \in F$, $c\mathbf{u} \in W$.

Question 9

Define Linear Combination.

Accepted Answer

A vector $\mathbf{v}$ is a linear combination of vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ if $\mathbf{v} = c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k$ for some scalars $c_1, \dots, c_k \in F$.

Question 10

What is the Span of a Set of Vectors $S = \{\mathbf{v}_1, \dots, \mathbf{v}_k\}$?

Accepted Answer

The set of all possible linear combinations of the vectors in $S$, denoted as $Span(S)$. It forms a subspace of the vector space $V$ containing $S$.

Question 11

Define Linear Independence.

Accepted Answer

A set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$ is linearly independent if the only solution to the vector equation $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ is the trivial solution $c_1 = c_2 = \dots = c_k = 0$.

Question 12

Define Linear Dependence.

Accepted Answer

A set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$ is linearly dependent if there exist scalars $c_1, \dots, c_k$, not all zero, such that $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$. This implies at least one vector can be written as a linear combination of the others.

Question 13

What is a Basis of a Vector Space $V$?

Accepted Answer

A set of vectors $B = \{\mathbf{b}_1, \dots, \mathbf{b}_n\}$ in $V$ that satisfies two conditions: 
1. $B$ is linearly independent. 
2. $B$ spans $V$ ($Span(B) = V$).

Question 14

What is the Dimension of a Vector Space $V$?

Accepted Answer

The number of vectors in any basis for $V$. If $V = \{\mathbf{0}\}$ (the zero vector space), its dimension is 0. If $V$ cannot be spanned by a finite set of vectors, it is infinite-dimensional.

Question 15

Define a Linear Transformation $T: V 	o W$.

Accepted Answer

A function $T$ from a vector space $V$ (domain) to a vector space $W$ (codomain) that satisfies two properties for all vectors $\mathbf{u}, \mathbf{v} \in V$ and scalar $c \in F$: 
1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$. 
2. Homogeneity: $T(c\mathbf{u}) = cT(\mathbf{u})$.

Question 16

What is the Kernel (Null Space) of a Linear Transformation $T: V 	o W$?

Accepted Answer

The set of all vectors $\mathbf{v}$ in $V$ such that $T(\mathbf{v}) = \mathbf{0}_W$, where $\mathbf{0}_W$ is the zero vector in $W$. Denoted as $Ker(T)$ or $Nul(T)$. $Ker(T) = \{\mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0}_W\}$. The kernel is a subspace of $V$.

Question 17

What is the Image (Range or Column Space) of a Linear Transformation $T: V 	o W$?

Accepted Answer

The set of all vectors $\mathbf{w}$ in $W$ that are images of at least one vector $\mathbf{v}$ in $V$. Denoted as $Im(T)$ or $Col(T)$. $Im(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}$. The image is a subspace of $W$.

Question 18

Define the Rank of a Linear Transformation $T$.

Accepted Answer

The dimension of the image (column space) of $T$, i.e., $rank(T) = dim(Im(T))$.

Question 19

Define the Nullity of a Linear Transformation $T$.

Accepted Answer

The dimension of the kernel (null space) of $T$, i.e., $nullity(T) = dim(Ker(T))$.

Question 20

State the Rank-Nullity Theorem.

Accepted Answer

For a linear transformation $T: V 	o W$, where $V$ is a finite-dimensional vector space, the sum of the rank and nullity of $T$ is equal to the dimension of the domain $V$. Mathematically: $rank(T) + nullity(T) = dim(V)$.

Question 21

How does the Rank-Nullity Theorem relate to matrices?

Accepted Answer

For an $m 	imes n$ matrix $A$, which represents a linear transformation $T(\mathbf{x}) = A\mathbf{x}$, the theorem states $rank(A) + nullity(A) = n$. Here, $rank(A)$ is the dimension of the column space of $A$, and $nullity(A)$ is the dimension of the null space of $A$ (which is the kernel of $T$).

Linear Algebra - Vector Spaces & Transformations

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