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?
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Under what conditions is W a subspace of V?
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What defines a linear combination of vectors?
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What does the span of a set S represent?
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Span S is a subspace of V.
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What does it mean for vectors to be linearly independent?
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What condition indicates that a set of vectors is linearly independent regarding the determinant?
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If S1 is linearly dependent, what can we say about S2 if S1 is a subset of S2?
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What does it mean for the vectors to form a basis for a vector space V?
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What are the components of the natural/standard basis for R3?
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What does it mean that every vector in V can be expressed as a unique linear combination of the basis vectors?
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What does it mean for a subset of vectors to serve as a basis for W?
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What is the dimension of a vector space V?
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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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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.