Vectors, Linear Combination, Span, and Basis Vectors Flashcards
(32 cards)
What is a vector?
A vector is a quantity that has both magnitude and direction.
True or False: A vector can be represented as an ordered pair of numbers in two-dimensional space.
True
Fill in the blank: The __________ of a vector is its length.
magnitude
What is a linear combination?
A linear combination is an expression made up of a set of vectors, where each vector is multiplied by a scalar and then summed.
Which of the following is a linear combination of vectors v1 and v2? A) 2v1 + 3v2 B) v1^2 + v2 C) 5v1 - v2
A) 2v1 + 3v2 and C) 5v1 - v2
What does the span of a set of vectors represent?
The span of a set of vectors is the set of all possible linear combinations of those vectors.
True or False: The span of a single non-zero vector is a line through the origin.
True
What is a basis vector?
A basis vector is a vector in a vector space that, when combined with other basis vectors, can represent any vector in that space.
How many vectors are needed to form a basis for a 3-dimensional space?
Three linearly independent vectors.
Fill in the blank: Two vectors are __________ if one can be expressed as a scalar multiple of the other.
linearly dependent
What is the relationship between the dimension of a vector space and its basis?
The dimension of a vector space is the number of vectors in any basis for that space.
True or False: Any set of vectors that spans a space is also a basis for that space.
False
What is the zero vector?
The zero vector is a vector with all its components equal to zero, and it does not have a direction.
Define orthogonal vectors.
Orthogonal vectors are vectors that are perpendicular to each other, meaning their dot product is zero.
What does it mean for vectors to be linearly independent?
Vectors are linearly independent if no vector in the set can be written as a linear combination of the others.
Fill in the blank: The __________ of a vector is often represented by the symbol ||v||.
magnitude
What is the standard basis for R^2?
The standard basis for R^2 is { (1, 0), (0, 1) }.
True or False: A basis for a vector space must consist of exactly the same number of vectors as the dimension of the space.
True
What is the geometric interpretation of the span of two vectors in R^2?
The span of two vectors in R^2 is the entire plane if the vectors are not collinear.
What is the dot product of two vectors?
The dot product is a scalar value obtained by multiplying corresponding components of two vectors and summing the results.
Fill in the blank: The __________ of a set of vectors must include only linearly independent vectors to form a basis.
basis
What is a coordinate vector?
A coordinate vector is a representation of a vector in terms of the basis vectors of a vector space.
True or False: The span of a set of vectors can be a subspace of a vector space.
True
What is the significance of the dimension of a vector space?
The dimension indicates the number of basis vectors required to span the space.
