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Linear Algebra
Linear Algebra Exam 1
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Terms in this set (24)
When is the vector unique?
when there are no free variables
Let A be a
7×7
matrix. What must a and b be in order to define
T : ℝ^a→ℝ^b
by
T(x)=Ax?
a= 7
b=7
When is a system consistent?
a linear system is consistent if and only if the right most column of the augmented matrix is not a pivot column, that is, if and only if an echelon form of the augmented matrix has no row of the form [0 ... 0 c] with c nonzero.
Notice that none of the rows in the row echelon form of the augmented matrix is of the form [0 ... 0 c]. Therefore, the system represented by the matrix is consistent.
When is there a nontrivial solution?
The homogeneous equation
Ax=0
has a nontrivial solution if and only if the equation has at least one free variable
Because column 3 of the echelon form of A0 is not a pivot column, it means that x3 is a free variable.
Since x3 is a free variable, the system has a nontrivial solution.
Reduced Echelon Form
reduced echelon form because the leading entry in each nonzero row is 1 and each leading 1 is the only nonzero entry in its column.
a. A homogeneous equation is always consistent.
True. A homogenous equation can be written in the form Ax=0,
where A is an m×n matrix and 0 is the zero vector in
ℝm. Such a system Ax=0 always has at least one solution, namely, x=0.
Thus a homogenous equation is always consistent.
The equation Ax=0 gives an explicit description of its solution set.?
False. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set.
The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.?
False. The homogeneous equation
Ax=0 always has the trivial solution.
The equation x=p+tv describes a line through v parallel to p.
False. The effect of adding p to v is to move v in a direction parallel to the line through p and
0.
So the equation
x=p+tv
describes a line through p parallel to
v.
The solution set of Ax=b
is the set of all vectors of the form w=p+vh, where
vh is any solution of the equation Ax=0.
False. The solution set could be empty. The statement is only true when the equation
Ax=bis consistent for some given b,
and there exists a vector p such that p is a solution.
A homogeneous system of equations can be inconsistent.
False. A homogeneous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm.Such a system Ax=0 always has at least onesolution, namely x=0.
Thus, a homogeneous system of equations cannot be inconsistent.
If x is a nontrivial solution of
Ax=0,
then every entry in x is nonzero. Choose the correct answer below
False. A nontrivial solution of Ax=0 is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero
The effect of adding p to a vector is to move the vector in a direction parallel to p.
True. Given v and p in ℝ2 or ℝ3,
the effect of adding p to v is to move v in a direction parallel to the line through p and
0.
The equation
Ax=b
is homogeneous if the zero vector is a solution
True. A system of linear equations is said to be homogeneous if it can be written in the form
Ax=0,
where A is an
m×n
matrix and 0 is the zero vector in
ℝm.
If the zero vector is a solution, then
b=Ax=A0=0.
If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0. Choose the correct answer below.
True. Suppose the equation
Ax=b
is consistent for some given
b,
and let p be a solution. Then the solution set of
Ax=b
is the set of all vectors of the form
w=p+vh,
where
vh
is any solution of the homogeneous equation
Ax=0
Suppose
Ax=b
has a solution. Explain why the solution is unique precisely when
Ax=0
has only the trivial solution.
Since Ax=b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax=b is a single vector if and only if the solution set of Ax=0is a single vector, and that happens if and only if
Ax=0
has only the trivial solution
If
b≠0,
can the solution set of
Ax=b
be a plane through the origin?
No. If the solution set of Ax=b contained the origin, then 0 would satisfy A0=b, which is not true since b is not the zero vector.
Your answer is correct.
A is a 3×3 matrix with three pivot positions.
(a) Does the equation Ax=0 have a nontrivial solution?
(b) Does the equation Ax=b have at least one solution for every possible b?
a)no
b)yes
Let A be a 3×3
matrix with two pivot positions.
a) Does the equation
Ax=0
have a nontrivial solution?
b) Does the equation
Ax=b
have at least one solution for every possible
b?
a) Yes. Since A has 2 pivots, there is one free variable. So
Ax=0
has a nontrivial solution.
b) No. A has one free variable. To have at least one solution for every possible
b,
A cannot have any free variable.
vSuppose A is a 3×3 matrix and y is a vector in ℝ3 such that the equation Ax=y does not have a solution. Does there exist a vector z in ℝ3 such that the equation Ax=z has a unique solution?
No. Since
Ax=y
has no solution, then A cannot have a pivot in every row. So the equation
Ax=z
has at most two basic variables and at least one free variable for any
z.
Thus the solution set for
Ax=z
is either empty or has infinitely many elements
Determine if the vectors are linearly independent. Justify your answer.
[3 ;0; 0;] [9; 3; −12;] [6; 12; -24;]
The vector equation has
only the trivial solution, so the vectors
are linearly independent
Determine if the vectors are linearly independent. v1= [5; -2} v2= [-5;2]
The vectors are not linearly independent because if
c1=11 and c2=1,
both not zero, then
c1v1+c2v2=0.
An indexed set of vectors
in ℝn
is said to be linearly independent if the vector equation
x1v1+x2v2+...+xpvp=0 has only the trivial solution. The set
{v1,..., vp}
is said to be linearly dependent if there exist weights
c1, ..., cp,
not all zero
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
the augmented matrix
represents the equation Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0 has only the trivial solution. Therefore, the columns of A form a linearly independent set.
Determine if the columns of the matrix form a linearly independent set.
The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector
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