Weekly true / false quiz (closes at 2:00pm, Saturday 21 October 2017)
[10 marks total.]
– Vector spaces.
The vector spaces in this question, and their operations, are all defined in Example 2.1.
Determine which of the following statements are true.
In the vector space over the field , an example of the addition operation is
In the vector space over the field , an example of the scalar multiplication operation is
The polynomial is an element in the vector space over the field .
In the vector space over , the addition of matrices is defined to be by matrix multiplication.
– Linear combinations
Recall that if is a vector space, a linear combination of the vectors is any vector of the form where are elements of the field of .
Determine which of the following statements are true.
In the vector space over the field , the vector is a linear combination of and .
In the vector space over the field , the vector is a linear combination of and .
In the vector space over the field , the vector is a linear combination of and .
In the vector space over the field , any vector can be written as a linear combination of and .
– Span
Recall that the span of a sequence of vectors in a vector space is the set of all possible linear combinations of that sequence.
Determine which of the following statements are true.
If then .
If and then is a linear combination of .
In , the span of contains infinitely many elements.
If is such that , then .
– Linear independence
Recall that a sequence of vectors in a vector space is linearly independent when it is not possible to write any of the vectors in the sequences as a linear combination of the other vectors. So also Theorem 2.20 for a more useful equivalent formulation.
Determine which of the following statements are true.
If is linearly independent and is linearly independent, then is a linearly independent sequence,
If is a linearly independent sequence, then is linearly independent,
The sequence of three vectors , form a linearly independent sequence in the vector space .
It is impossible to have a sequence of two vectors in which is linearly independent.
– Basis
For each of the following sequences of vectors, determine which ones form a basis of the subspace of upper-triangular 2 by 2 real matrices.
is a basis of ,
is a basis of ,
is a basis of ,
is a basis of .