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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
What is the initial value for x cubed?
X (or any letter) is a variable; you can not know the initial value unless the question specifically states what it is.
It can be manipulated into varying written forms; but the X notation itself will remain a variable.
When given a variable in the form of a letter; the only thing you can typically assume is that it is different from the other letters in the equation in some way or it would likely just have the same letter used multiple times throughout. (though this is not always an accurate assumption, there are cases where different letters end up being the same number, they just were not known at the start of the equation and are thus being represented by different variable letters)
Alternate written methods:
X to the third power = X^3 = (X)(X)(X)
What is a symmetric determinant?
A determinant D = dij where i = 1,2,...,n and j = 1,2,...,n is symmetric if dij = dji for all i, j.
A determinant D = dij where i = 1,2,...,n and j = 1,2,...,n is symmetric if dij = dji for all i, j.
A determinant D = dij where i = 1,2,...,n and j = 1,2,...,n is symmetric if dij = dji for all i, j.
A determinant D = dij where i = 1,2,...,n and j = 1,2,...,n is symmetric if dij = dji for all i, j.
What is the boolean expression for a 2 out of 3 OR gate?
If the inputs are ABC. The inputs required to give an output are ABC, AB, AC and BC. Using the Absorption law X + X.Y = X we can remove ABC, the inputs required are therefore AB, AC and BC.
What If a system of linear equations has no solution?
It can happen. Then there is no solution!
It can happen. Then there is no solution!
It can happen. Then there is no solution!
It can happen. Then there is no solution!
When solving a system of linear inequalities what does the region that is never shaded represent?
It represents the solution set.
What is a two step equation that equals 28?
.5x+3=17 subtract 3 from each side and multiply both sides by 2 and you have x=28
What is the purpose of locating the circumcenter?
It identifies the point which is equidistant from the three vertices of the triangle. If, for example, you wanted to locate a phone mast to supply three remote locations, a good starting point would be to find somewhere which is at or very close to the circumcentre. Of course, local geography (hills, for example) would have to be taken into account.
3x - 12 = 10x - 15 - 6x -10
3x - 12 = 4x -25
3x + 13 = 4x
13 = 4x - 3x
13 = x
None?
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.
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.
What is the definition of Simultaneous Linear Equations?
A system of linear equations is two or more simultaneous linear equations. In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
How many quarters and dimes in 1.95?
7 quarters equal $1.75
2 dimes equal $.20
You can also get it with 5 quarters and 7 dimes.
3 quarters and 12 dimes and one other way.
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
