LINEAR ALGEBRA EXAM Q&A
Triangular System Definition - Answer-1) Each column variable is pivot in exactly one
equation
2) Same # of rows as columns
3) exactly one solution
Echelon form - Answer-Each pivot has only zeros below it
Gaussian Elimination - Answer-How to convert matrix to echelon form
Reduced Row Echelon Form (RREF) - Answer-In echelon form &
1) All pivots contain a 1
2) 0's above and below pivots
Homogenous Equation - Answer-Has form:
a1x1 + a2x2... = 0
Always consistent, since trivial solution (each variable =0 ) is possible
Diagonal Dominance - Answer-Means that coefficients on diagonal are >= sum of other
coefficients in row
Jacobi Iteration v Gauss Seidel Iteration - Answer-Jacobi is more basic. Gauss Seidel is
more advanced.
Jacobi:
Step 1) Set row of system/matrix in terms of a single variable
Want it to like this:
x1 = 1 + x2
x2 = 3 + x3
x3 = 5 + x1
Step 2) Set all the variables on right hand side of = 0.
So:
x1 = 1 + 0
, x2 = 3 + 0
x3 = 5 + 0
Step 3:
Now take the value we solved for and plug them in:
So:
x1 = 1 + 3 : x1 = 4
x2 = 3 + 5 : x2 = 6
x3 = 5 + 1: x3 = 6
S4: Now plug hese new values back into the original equations.
Keep iterating. If Matrix is diagonally dominant this should converge to answers.
Gauss Seidel
We only set two of the variables = 0, Solve them first and then plug in.
Diagonally Dominant - Answer-Matrix is diagonally dominant if the sum every row in
matrix off the diagonal is <= values on the diagonal
Span (what does it mean how to test if set m of vectors spans Rn) - Answer-Test:
1) If m is < n, immediately reject, otherwise proceed
2) Form matrix out of the m vectors. Transform to echelon form.
3) If every row has a pivot, then vectors span Rn.
Given Ax = b, if A is 3x4 matrix what will b look like - Answer-3x1 vector.
For Ax = b to be valid, x must: - Answer-x (i.e. vector of x variables) must have same
number of rows as A has columns
linear independence conceptual definition - Answer-set of vectors is linearly
independent if the only combination of variables (i.e. x1, x2...) which combine to make
the sum of the vectors =0 is "trivial solution (i.e. all the variables = 0
How to determine if set of vectors is linearly independent? - Answer-Create augmented
matrix containing all the vectors, with farthest right column set to all 0s.
What is domain? - Answer-Possible inputs in linear transformation (i.e. R2, R3...)
What is Codomain - Answer-"Space" (i.e. R2, R3...) where output vectors exist in
What is image? - Answer-The image u under T is a vector T(u) of output, given a vector
u of input, with T as a linear transformation
Triangular System Definition - Answer-1) Each column variable is pivot in exactly one
equation
2) Same # of rows as columns
3) exactly one solution
Echelon form - Answer-Each pivot has only zeros below it
Gaussian Elimination - Answer-How to convert matrix to echelon form
Reduced Row Echelon Form (RREF) - Answer-In echelon form &
1) All pivots contain a 1
2) 0's above and below pivots
Homogenous Equation - Answer-Has form:
a1x1 + a2x2... = 0
Always consistent, since trivial solution (each variable =0 ) is possible
Diagonal Dominance - Answer-Means that coefficients on diagonal are >= sum of other
coefficients in row
Jacobi Iteration v Gauss Seidel Iteration - Answer-Jacobi is more basic. Gauss Seidel is
more advanced.
Jacobi:
Step 1) Set row of system/matrix in terms of a single variable
Want it to like this:
x1 = 1 + x2
x2 = 3 + x3
x3 = 5 + x1
Step 2) Set all the variables on right hand side of = 0.
So:
x1 = 1 + 0
, x2 = 3 + 0
x3 = 5 + 0
Step 3:
Now take the value we solved for and plug them in:
So:
x1 = 1 + 3 : x1 = 4
x2 = 3 + 5 : x2 = 6
x3 = 5 + 1: x3 = 6
S4: Now plug hese new values back into the original equations.
Keep iterating. If Matrix is diagonally dominant this should converge to answers.
Gauss Seidel
We only set two of the variables = 0, Solve them first and then plug in.
Diagonally Dominant - Answer-Matrix is diagonally dominant if the sum every row in
matrix off the diagonal is <= values on the diagonal
Span (what does it mean how to test if set m of vectors spans Rn) - Answer-Test:
1) If m is < n, immediately reject, otherwise proceed
2) Form matrix out of the m vectors. Transform to echelon form.
3) If every row has a pivot, then vectors span Rn.
Given Ax = b, if A is 3x4 matrix what will b look like - Answer-3x1 vector.
For Ax = b to be valid, x must: - Answer-x (i.e. vector of x variables) must have same
number of rows as A has columns
linear independence conceptual definition - Answer-set of vectors is linearly
independent if the only combination of variables (i.e. x1, x2...) which combine to make
the sum of the vectors =0 is "trivial solution (i.e. all the variables = 0
How to determine if set of vectors is linearly independent? - Answer-Create augmented
matrix containing all the vectors, with farthest right column set to all 0s.
What is domain? - Answer-Possible inputs in linear transformation (i.e. R2, R3...)
What is Codomain - Answer-"Space" (i.e. R2, R3...) where output vectors exist in
What is image? - Answer-The image u under T is a vector T(u) of output, given a vector
u of input, with T as a linear transformation
