GATE MATHEMATICS 2020 | Linear Algebra | Linear Transformation

Hello friends, this is Mahesh Yuvraj and you are watching Next Gen Maths. If you have not subscribed my channel, then please subscribe the channel and press the bell icon of Next Gen Maths. In this video, we are going to discuss the problem of Gate 20-20 and problem of Linear Algebra. Example, suppose that T r4 to rx is a linear transformation over R satisfying T of minus 1, 1, 1, 1 equal to x square plus 2 x power 4, T of 1, 2, 3, 4 equal to 1 minus x square and T of 2 minus 1, minus 1, 0 equal to x cube minus x power 4, then the coefficient of x power 4 in T of minus 3, 5, 6, 6 is. If you want notes of this solution, then please visit our website www.nextgenmaths.com, in the description link is available. Now it is a problem of linear transformation and we have given the image of 3 vector and using this 3 vector image, we have to find the image of 4th vector. So we are going to use the property of linear transformation, if z equal to alpha x plus beta y, then T of z equal to alpha times T x plus beta times T y. Now we have given 3 vector v1, v2, v3 and we are going to find the image of 4th vector v4. So first we try to write down 4th vector as a linear combination of first 2 vector, that means v4 equal to alpha times v1 plus beta times v2 and gamma times v3 and we try to find the value of alpha, beta, gamma from this vector equation, just we simplify and we will have 4 linear equation from this vector equation, first equation is minus alpha plus beta plus 2 gamma equal to minus 3, alpha plus 2 beta minus gamma equal to 5, alpha plus 3 beta minus gamma equal to 6 and alpha plus 4 beta equal to 6. Now we have to find the value of alpha, beta and gamma, for that first we are doing 3 minus 2, then immediately we will get the value of beta equal to 1. Use equation 4 to find the value of alpha, if beta equal to 1, then alpha equal to 6 minus 4 equal to 2, now use the value of beta and alpha and substitute in the equation 2, immediately we will get the value of gamma equal to minus 1. Therefore v4 equal to alpha times v1 plus beta times v2 plus gamma times v3 equal to 2 times v1 plus 1 times v2 minus 1 times v3. Now use the property of linear transformation, so T of v4 equal to 2 times T v1 plus T v2 minus T v3, now we have a image of v1, v2, v3 with respect to T, so we will use that image and substitute that image, so image of v1 that means minus 1, 1, 1 is x square plus 2 x power 4, image of v2 is 1 minus x square and image of v3 is x cube minus x power 4. Now if we simplify this term, then we have 5 x power 4 minus x cube plus x square plus 1 and it is the image of v4 that means T of minus 3, 5, 6, 6 and we want to find the coefficient of x power 4 in the image of v4. And its coefficient is 5, your answer is 5. Thanks for watching, for more videos like share, subscribe, press the bell icon of NextGenMath.