Here is an augmented matrix in which * denotes an arbitrary number and # denotes a
nonzero number. (10%)
Which of the following statements is (are) true?
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The given augmented matrix is consistent and the solution is unique.
The given augmented matrix is inconsistent and the solution is unique.
The given augmented matrix is consistent and the solution is not unique.
The given augmented matrix is inconsistent and the solution is not unique.
None of the above.
Assume A is an m× n matrix. If rank A = n , then
The columns of A are linearly independent
Ax = b has at least one solution for every b in Rm .
Every column of its reduced row echelon form contains a pivot position.
Ax = b has at most one solution for every b in Rm .
None of the above statements is true.
For the four vectors below, which of the following statements is (are) true?
They are linearly independent.
They span R3 .
They are linearly dependent.
They do not span R3 .
None of the above.
Let V be a subspace of R n with dimension k . Which of the following statements
is (are) true?
Every linearly independent subset of V contains at least k vectors.
Any finite subset of V containing more than k vectors is linearly dependent.
n ≥ k .
Any finite subset of V containing less than k vectors is linearly independent.
None of the above.
Suppose that s, t, and u are vectors in Rn such that s is orthogonal to u and
u is orthogonal to t . Then
s is orthogonal to t
For any orthogonal n× n matrix P , we have that Pu is orthogonal to both s
and t
For any orthogonal n× n matrix P , we have that Ps is orthogonal to u
s + t is orthogonal to u
None of the preceding statements are true
Let M be an inner product space
Since the inner product is an integral of a product of functions, M is a function
space
c u, v = cu, cv for all vectors u and v in M and for every scalar c
T(u), v = u, T(v) for all vectors u and v in M and for every linear
operator T on M
If M is finite-dimensional, then M contains an orthonormal basis
None of the preceding statements are true.
An m× n matrix P is invertible if
The reduced row echelon form of P is In
The columns of P are linearly independent
The columns of P span Rm
The rows of P are linearly independent
None of the preceding statements is true.
Suppose that S is an arbitrary n× n matrix. Then
det S is the sum of its diagonal entries
det S2 = 2 det S
det S is a vector in Rn
det S = det ST
None of the preceding statements is true.
Let A be a subset of R³ containing two or more vectors. Then:
The span of any two vectors in A is a plane in R3
If A contains more than three vectors, then A is linearly independent
The span of any two nonzero vectors in A is a plane in R3
Every vector in A is in the span of A
None of the preceding statements is true.
The solution for f (t) of the following equation
k1 + k2 = 3
k2 + g(0) = 1
g(1) + g(0) = 4
g(1) + g(−1) = 8
none of above.
We are going to solve of the following differential equation system:
with initial values as x1(0) = 4 , x2 (0) = 7 and x3 (0) = 7 . The solution has the form as
where |λ1 |≤|λ2 |≤|λ3 | and Aij s are constants.
Which of the following item(s) is(are) true:
λ1 +λ2 = 3
2λ1 +λ3 = 3
λ2 +λ3 = 3
λ1 +λ3 = 3
none of above.
We are going to solve of the following differential equation system:
with initial values as x1(0) = 4 , x2 (0) = 7 and x3 (0) = 7 . The solution has the form as
where |λ1 |≤|λ2 |≤|λ3 | and Aij s are constants.
Which of the following item(s) is(are) true:
x1(1) + 2x2 (1) = 6e
x1(1) + x3 (1) = 2cos(1)
x2 (1) + x3 (1) = 8e
x1(1) − x2 (1) = 2cos(1)
none of above.
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