MAT2041 Assignment 4: Linear space, Basis, Dimension, Solving Ax=b
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MAT2041
Assignment 4: Linear space, Basis, Dimension, Solving Ax=b
1. (Judgement) (5 × 2 = 10 points)
Judge whether each statement is true or false. If true, provide a brief explanation why it is true. If false, provide a counter-example (i.e., a specific example that the statement does not hold)
(a) The symmetric matrices (i.e. those matrices satisfying A = A⊤) with dimension n × n form a subspace of Rn×n .
(b) The unsymmetric matrices (i.e. those matrices satisfying A 
A⊤) with dimen- sion n × n form a subspace of Rn×n .
(c) The singular matrices with dimension n × n (i.e. the matrices that are not invertible) form a subspace of Rn×n .
(d) Suppose W is a subspace of R5 and dim(W) = 2. Then any three non-zero vectors in W must be linearly dependent.
(e) If the columns of a matrix are linearly dependent, then the rows of this matrix must also be linearly dependent.
2. Check whether the following vectors are linearly independent or dependent. If they are linearly dependent, find one vector which can be expressed by remaining vectors and find its expression. (2 × 5 = 10 points)
(a)
\ 0(3) ) \ −2(4) ) \5(2) )
α 1 = 图(图) 2 图(图) , α2 = 图(图) 1 图(图) , α3 = 图(图) 0 图(图)
( ) ( ) ( )
(b)
\ −1 ) \ 4 ) \ 6 ) \ 3 )
( 0 ) ( −3 ) ( −2 ) ( 1 )
3. Calculate the rank of following matrix. ( 10 points)
\ 
)
( 
)
4. Find the solution set of following linear systems. (4 + 3 + 3 = 10 points)
(i)
( x1 + 2x2 − 3x3 − 4x4 = −5
,
,
,
,
( −9x1 − 4x2 − x3 = 17
(ii) The linear system corresponding to the following augmented matrix
」'
'「0 0 0 1 36 | 8l(')
(iii) The linear system corresponding to the following augmented matrix
」'
'(')0 0 0 0 1 5 0 8 | 3'(')
l(')
5. Please prove that if vectors α 1 , α2 , α3 are linearly independent, then vectors 3α1 − α2 , 5α2 + 2α3 , 4α3 − 7α1 are also linearly independent. ( 10 points)
6. Let A be an 8 × n matrix of rank T . For each pair of values of T and n below, how many solutions could one have for the linear system Ax = b? Explain your answers. ( 2.5 × 4 = 10 points)
(a) n = 7, T = 7.
(b) n = 9, T = 7;
(c) n = 9, T = 8;
(d) n = 8, r = 8.
7. Let A ∈ Rm×n be an arbitrary matrix, B ∈ Rn×n be a square matrix. Prove that (2 × 5 = 10 points)
(a) rank(AB) ≤ rank(A);
(b) If rank(B) = n, then rank(AB) = rank(A).
8. Please prove that, for any positive integer n, the functions 1, cos x, cos 2x, cos 3x, ..., cos nx are linearly independent. ( 10 points)
9. Let r < n. In Rn, we let
U = {(a1 , a2 , · · · , ar , 0, · · · , 0)⊤ } | ai ∈ R, i = 1, 2, · · · , r }
Find the basis and dimension of the subspace U . ( 10 points)
10. Show that if the product of two matrices is the zero matrix, AB = 0, then the column space of B is contained in the nullspace of A. What about the row space of A and
the left nullspace of B ? Can you make a similar statement?
( 10 points)
2022-11-07