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Linear Algebra
Linear algebra is the detailed study of vector spaces. With applications in such disparate fields as sociology, economics, computer programming, chemistry, and physics, including its essential role in mathematically describing quantum mechanics and the theory of relativity, linear algebra has become one of the most essential mathematical disciplines for the modern world. Please direct all questions regarding matrices, determinants, eigenvalues, eigenvectors, and linear transformations into this category.
2,176 Questions
How do you use the substitution method to solve a system of linear equations?
You first need to isolate one variable in one of the equations and then substitute that value into the other equation and solve for the remaining variable. Take the value you just got and plug it in to the other equation for the appropriate variable. Solve for the first variable that you isolated.
Example: 2x-y=3, 4x+2y=6
2x-y=3 Isolate y.
+y +y
2x=y+3
-3 -3
y=2x-3 If y=2x-3 then substitute 2x-3 in for y in the other equation.
4x+2y=6
4x+2(2x-3)=6 Distribute
4x+4x-6=6 Simplify
8x-6=6
+6 +6
8x=12 Divide both sides by 8 to isolate x.
x=12/8 Simplify
x=3/2 Now substitute 3/2 in for x in the first equation.
2x-y=3
2(3/2)-y=3 Again, distribute.
6/2-y=3 Simplify
3-y=3 Isolate y.
-3 -3
y=0 (It can't be -y=0, because you can't have -0)
So, x=3/2 and y=0!
What two numbers multiply to get 200 but add to get 20?
The factor pairs of 200 are (200,1)(100,2)(50,4)(40,5)(25,8)(20,10) None of them have a sum or difference of 20.
Factors of 200 that add to 20 are 10 and 10.
What are the applications of cramer's rule?
Cramer's rule is applied to obtain the solution when a system of n linear equations in n variables has a unique solution.
3x - 12 = 10x - 15 - 6x -10
3x - 12 = 4x -25
3x + 13 = 4x
13 = 4x - 3x
13 = x
None?
Can you multiply a 2x2 matrices?
You can definitely multiply 2x2 matrices with each other. In fact you can multiply a AxB matrix with a BxC matrix, where A, B, and C are natural numbers. That is, the number of columns of the first matrix must equal the number of rows of the second matrix--we call this "inner dimensions must match."
How can you tell from a linear equation which direction it goes without graphing the equation?
You must find the slope, if it is positive, then the line is always increasing.
If it is negative, then the line is always decreasing.
What graph indicates that there is a linear relationship?
A straight line graph plotted on the Cartesian plane
How many scores in 10 decades?
A score is 20 years. A decade is 10 years. So in 10 decades we've had 100 years. Thus there are 5 scores in 10 decades.
10 Years/1 Decade x 10 Decades ÷ 20 years/1 score = 5 scores
How do you solve 3x-y1 9x-3y equals -3?
It is not possible to give a sensible answer because the operator between y1 and 9x is not visible.
How many millaliters is 300 marbles?
None, since there is no such thing as a millaliter.
Furthermore, marbles come in all sorts of sizes.
x and y are complementary so x + y = 90 and so y = 90 - x
z and q are complementary so z + q = 90 and so q = 90 - z
x = z
so 90 - x = 90 - z
that is y = q
How can you tell if a matrix is invertible?
An easy exclusion criterion is a matrix that is not nxn. Only a square matrices are invertible (have an inverse). For the matrix to be invertible, the vectors (as columns) must be linearly independent. In other words, you have to check that for an nxn matrix given by {v1 v2 v3 ••• vn} with n vectors with n components, there are not constants (a, b, c, etc) not all zero such that av1 + bv2 + cv3 + ••• + kvn = 0 (meaning only the trivial solution of a=b=c=k=0 works).
So all you're doing is making sure that the vectors of your matrix are linearly independent. The matrix is invertible if and only if the vectors are linearly independent. Making sure the only solution is the trivial case can be quite involved, and you don't want to do this for large matrices. Therefore, an alternative method is to just make sure the determinant is not 0. Remember that the vectors of a matrix "A" are linearly independent if and only if detA�0, and by the same token, a matrix "A" is invertible if and only if detA�0.
What does it means if there are an infinite number of solutions to a system of equations?
In simple terms all that it means that there are more solutions than you can count!
If the equations are all linear, some possibilities are given below (some are equivalent statements):
- there are fewer equations than variables
- the matrix of coefficients is singular
- the matrix of coefficients cannot be inverted
- one of the equations is a linear combination of the others
Can you solve system of equations by graphing?
Yes you can, if the solution or solutions is/are real.
-- Draw the graphs of both equations on the same coordinate space on the
same piece of graph paper.
-- Any point that's on both graphs, i.e. where they cross, is a solution of the
system of equations.
-- If both equations are linear, then there can't be more than one such point.
How would you determine the slope of a linear function from a table?
slope = change in y values divided by change in x values.
m = (y2-y1)/ (x2-x1)
pick 2 ordered pairs from the table and use the formula above.
How an algebric expression with quadratic denominator is solved?
Algebraic expressions can't be solved because they are not equations but they can be simplified.
