LINEAR ALGEBRA FINAL EXAM Q&A
A system of equations is consistent if - Answer-it has only one solution
We say that the set of vectors S = {V1,V2,..,Vn} are linearly dependent if - Answer-we
can find a solution to X1V1+X2V2+..+XnVn = 0 where not all of the coefficients equal 0.
We say that the set of vectors S = {V1,V2,..,Vn} are linearly independent if - Answer-
whenever X1V1 + X2V2 +...+XnVn = 0 then X1=X2=...=Xn = 0
An eigenvector of an nxn matrix A is a nonzero vector x satisfying - Answer-A(x)=λx for
some scalar λ.
We say that the set of vectors S = {V1,V2,..,Vn} spans a vector space V if - Answer-
every vector v∈V is a linear combination of V1,V2,..,Vn
We say that vector V is a linear combination of the vectors V1,V2,..,Vn provided -
Answer-there exists scalars x1,x2,..xn satisfying V = x1v1 + x2v2+...+xnvn
A function T: R → R is a linear transformation if it satisfies - Answer-a. T(x+y) = T(x)
+T(y) for all vectors x,y∈R and
b. T(cx) = cT(x) for all vectors x∈R and for all scalars c∈R
The three elementary row operations are - Answer-a. Replace one row with the sum of
itself and a multiple of another row
b. Interchange 2 rows
c. Multiply all entries in the row by a nonzero constant
A rectangular matrix is in reduced echelon form if it has the following properties -
Answer-a. All nonzero rows are above any rows of all zeros
b. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
c. The leading entry in each nonzero is a 1.
d. Each leading 1 is the only nonzero entry in its column.
Let A be a nxn matrix A scalar λ is an eigenvalue of A if - Answer-A(x) =λx for some
nonzero vector x
If T is a linear transformation, T: R→R then the null space of T - Answer-Nul T = {x∈R:
T(x) =0}
A function f: x→y is onto if - Answer-for each y∈Y there is at least x∈X with f(x)=y
A function f: x→y is 1-1 if - Answer-whenever f(r)=f(s) then r=s
A system of equations is consistent if - Answer-it has only one solution
We say that the set of vectors S = {V1,V2,..,Vn} are linearly dependent if - Answer-we
can find a solution to X1V1+X2V2+..+XnVn = 0 where not all of the coefficients equal 0.
We say that the set of vectors S = {V1,V2,..,Vn} are linearly independent if - Answer-
whenever X1V1 + X2V2 +...+XnVn = 0 then X1=X2=...=Xn = 0
An eigenvector of an nxn matrix A is a nonzero vector x satisfying - Answer-A(x)=λx for
some scalar λ.
We say that the set of vectors S = {V1,V2,..,Vn} spans a vector space V if - Answer-
every vector v∈V is a linear combination of V1,V2,..,Vn
We say that vector V is a linear combination of the vectors V1,V2,..,Vn provided -
Answer-there exists scalars x1,x2,..xn satisfying V = x1v1 + x2v2+...+xnvn
A function T: R → R is a linear transformation if it satisfies - Answer-a. T(x+y) = T(x)
+T(y) for all vectors x,y∈R and
b. T(cx) = cT(x) for all vectors x∈R and for all scalars c∈R
The three elementary row operations are - Answer-a. Replace one row with the sum of
itself and a multiple of another row
b. Interchange 2 rows
c. Multiply all entries in the row by a nonzero constant
A rectangular matrix is in reduced echelon form if it has the following properties -
Answer-a. All nonzero rows are above any rows of all zeros
b. Each leading entry of a row is in a column to the right of the leading entry of the row
above it.
c. The leading entry in each nonzero is a 1.
d. Each leading 1 is the only nonzero entry in its column.
Let A be a nxn matrix A scalar λ is an eigenvalue of A if - Answer-A(x) =λx for some
nonzero vector x
If T is a linear transformation, T: R→R then the null space of T - Answer-Nul T = {x∈R:
T(x) =0}
A function f: x→y is onto if - Answer-for each y∈Y there is at least x∈X with f(x)=y
A function f: x→y is 1-1 if - Answer-whenever f(r)=f(s) then r=s
