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If we take one green x vector and two blue y vectors in gray , we get the red vector. The third way to look at this system entirely through matrices and use the matrix form of the equations. How to solve the equations written in matrix form will be discussed in the next lectures.

But I can tell you beforehand that the method is called Gauss elimination with back substitution.

MA251 Algebra 1: Advanced Linear Algebra

We can no longer plot it in two dimensions because there are three unknowns. This is going to be a 3D plot. Since the equations are linear in unknowns x, y, z, we are going to get three planes intersecting at a single point if there is a solution. Notice how difficult it is to spot the point of intersection?

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Almost impossible! And all this of going one dimension higher. Imagine what happens if we go 4 or higher dimensions. The column picture is almost as difficult to understand as the row picture.

Here it is for this system of 3 equations in 3 unknowns:. The first column 2, -1, 0 is red, the second column -1, 2, -3 is green, the fourth column 0, -1, 4 is blue, and the result 0, -1, 4 is gray. Again, it's pretty hard to visualize how to manipulate these vectors to produce the solution vector 0, -1, 4. But we are lucky in this particular example.

Notice that if we take none of red vector, none of green vector and one of blue vector, we get the gray vector! That is, we didn't even need red and green vectors! This is all still tricky, and gets much more complicated if we go to more equations with more unknowns.

Linear Algebra - Open Textbook Library

Therefore we need better methods for solving systems of equations than drawing plane or column pictures. They must also understand the equivalence of linear maps between vector spaces and matrices and be able to row reduce a matrix, compute its rank and solve systems of linear equations. The definition of a determinant in all dimensions will be given in detail, together with applications and techniques for calculating determinants.

Students must know the definition of the eigenvalues and eigenvectors of a linear map or matrix, and know how to calculate them. Archived Pages: Pre Year 3 regs and modules G G Year 4 regs and modules G Past Exams Core module averages. These applications include: Solutions of simultaneous linear equations. Unlike other parts of mathematics that are frequently invigorated by new ideas and unsolved problems, linear algebra is very well understood.

Its value lies in its many applications, from mathematical physics to modern algebra and coding theory. Linear algebra usually starts with the study of vectors, which are understood as quantities having both magnitude and direction.

Linear algebra

Vectors lend themselves readily to physical applications. For example, consider a solid object that is free to move in any direction. When two forces act at the same time on this object, they produce a combined effect that is the same as a single force. To picture this, represent the two forces v and w as arrows; the direction of each arrow gives the direction of the force, and its length gives the magnitude of the force.

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Vectors are often expressed using coordinates. For example, in two dimensions a vector can be defined by a pair of coordinates a 1 , a 2 describing an arrow going from the origin 0, 0 to the point a 1 , a 2.

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In three dimensions a vector is expressed using three coordinates a 1 , a 2 , a 3 , and this idea extends to any number of dimensions. Representing vectors as arrows in two or three dimensions is a starting point, but linear algebra has been applied in contexts where this is no longer appropriate. For example, in some types of differential equations the sum of two solutions gives a third solution, and any constant multiple of a solution is also a solution.

In such cases the solutions can be treated as vectors, and the set of solutions is a vector space in the following sense.

The Big Picture of Linear Algebra

The numbers are called scalars because in early examples they were ordinary numbers that altered the scale, or length, of a vector. For example, if v is a vector and 2 is a scalar, then 2 v is a vector in the same direction as v but twice as long.