MATH2051 - Linear Algebra II Flashcards

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Questions and Answers

What is a vector in the plane?

A vector in the plane is a 2x1 matrix x = [x y] where x and y are real numbers.

How is the sum of two vectors u and v represented?

The sum of the vectors u and v is represented as u + v = [u1 + v1, u2 + v2].

What is scalar multiplication in vector operations?

Scalar multiplication of a vector u = [u1, u2] by a scalar c results in c*u = [cu1, cu2].

What are the eight properties of a real vector space?

The properties include commutativity, associativity, identity element, inverse element for addition, and distributive properties for scalar multiplication. Signup and view all the answers

Under what conditions is W a subspace of V?

W is a subspace of V if it is non-empty and closed under addition and scalar multiplication. Signup and view all the answers

What defines a linear combination of vectors?

A vector v is a linear combination of vectors v1, v2,..., vk if v = a1v1 + a2v2 + ... + ak*vk for some real numbers a1, a2,..., ak. Signup and view all the answers

What does the span of a set S represent?

Span S is the set of all vectors in V that can be formed as linear combinations of the vectors in S. Signup and view all the answers

Span S is a subspace of V.

True Signup and view all the answers

What does it mean for vectors to be linearly independent?

Vectors are linearly independent if the only solution to the equation a1v1 + a2v2 + ... + ak*vk = 0 is a1 = a2 = ... = ak = 0. Signup and view all the answers

What condition indicates that a set of vectors is linearly independent regarding the determinant?

A set of vectors is linearly independent if the determinant of the matrix formed by the vectors is not equal to zero. Signup and view all the answers

If S1 is linearly dependent, what can we say about S2 if S1 is a subset of S2?

If S1 is linearly dependent, then S2 is also linearly dependent. Signup and view all the answers

What does it mean for the vectors to form a basis for a vector space V?

The vectors form a basis for V if they span V and are linearly independent. Signup and view all the answers

What are the components of the natural/standard basis for R3?

{[1, 0, 0], [0, 1, 0], [0, 0, 1]} Signup and view all the answers

What does it mean that every vector in V can be expressed as a unique linear combination of the basis vectors?

It means that if S is a basis for V, then each vector in V corresponds to one specific combination of vectors from S. Signup and view all the answers

What does it mean for a subset of vectors to serve as a basis for W?

A subset of S is a basis for W if it is linearly independent and spans W. Signup and view all the answers

What is the dimension of a vector space V?

The dimension of V is the number of vectors in a basis for V. Signup and view all the answers

Study Notes

Vectors and Operations

A vector in the plane is represented as a 2x1 matrix x=[x y], with x and y as real number components.
The sum of two vectors u = [u1 u2] and v = [v1 v2] results in u + v = [u1 + v1, u2 + v2].
Scalar multiplication of vector u = [u1 u2] by scalar c yields cu = [cu1, cu2], altering the direction if c < 0.

Properties of Vector Spaces

Vector addition is commutative and associative; adding the zero vector returns the original vector.
Each vector u has an additive inverse denoted by -u, satisfying u + (-u) = 0.
Scalar multiplication distributes over vector addition and respects scalar addition and multiplication rules.

Subspaces

A non-empty subset W of a vector space V is a subspace if it is closed under vector addition and scalar multiplication.

Linear Combinations and Spanning

A vector v in vector space V is a linear combination of vectors v1, v2,..., vk if v = a1v1 + a2v2 + ... + akvk for real coefficients a1, a2,..., ak.
The span of a set of vectors S consists of all possible linear combinations of vectors in S.
Span S is a subspace of V.

Linear Independence and Basis

Vectors v1, v2,..., vk are linearly dependent if a non-trivial linear combination equals zero, otherwise they are independent.
A set of vectors forms a basis for vector space V if they span V and are linearly independent.
The determinant of a matrix whose columns are vectors indicates linear independence; specifically, non-zero determinant implies independence.

Theorems on Linear Combinations

Linear combinations can reveal independence; if any vector in a set is dependent on the preceding vectors, then the entire set is dependent.
Any basis S of vector space V guarantees that every vector in V can be uniquely expressed as a linear combination of vectors from S.

Dimension

The dimension of a vector space V is defined as the count of vectors in any basis for V.

Standard Basis

For R^3, the standard basis is defined as {[1, 0, 0], [0, 1, 0], [0, 0, 1]}.
For polynomial space Pn, the standard basis consists of {t^n, t^(n-1), ..., t, 1}.

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Description

Test your knowledge of key concepts in Linear Algebra II with these flashcards. This set covers essential definitions regarding vectors in the plane, vector addition, and scalar multiplication. Ideal for students looking to reinforce their understanding of the subject.